If you're looking to kickstart your project, consider starting with something along these lines:

.goog-te-menu-frame {
    max-width:100% !important; //adjust as needed
}

Additionally, you might want to implement the following:

.goog-te-menu2 { //the container for options table
    max-width: 100% !important;
    overflow: scroll !important;
    box-sizing:border-box !important; //addresses padding
    height:auto !important; //eliminates vertical scrolling
}

Unfortunately, modifying the second part is challenging due to the translate interface being embedded in an iframe on your webpage. Luckily, since it shares the same domain, we can manipulate it using Javascript like this:

$('.goog-te-menu-frame').contents().find('.goog-te-menu2').css(
    {
        'max-width':'100%',
        'overflow':'scroll',
        'box-sizing':'border-box',
        'height':'auto'
    }
)

However, this solution won't work until the element actually loads (asynchronously), which requires a workaround that can be found here. To tie it all together, here's what you should do:

function changeGoogleStyles() {
    if($('.goog-te-menu-frame').contents().find('.goog-te-menu2').length) {
        $('.goog-te-menu-frame').contents().find('.goog-te-menu2').css(
            {
                'max-width':'100%',
                'overflow':'scroll',
                'box-sizing':'border-box',
                'height':'auto'
            }
        )
    } else {
        setTimeout(changeGoogleStyles, 50);
    }
}
changeGoogleStyles();

Phew.

You can apply a similar approach to customize other aspects of the translate box or tweak the table styles to ensure it flows vertically instead of extending horizontally off-screen. For further guidance, refer to this answer.

EDIT:

Even this doesn't suffice, given Google re-applies the styles every time you interact with the dropdown. While attempting to modify height and box-sizing, Google consistently overrides those changes, yet overflow and max-width persist. To safeguard our styles from being overridden, we must employ inline styles and include !important declarations [sigh]. Here's how you can achieve this (I've replaced the selector with a variable for brevity and potentially minimal performance enhancement):

function changeGoogleStyles() {
    if(($goog = $('.goog-te-menu-frame').contents().find('body')).length) {
        var stylesHtml = '<style>'+
            '.goog-te-menu2 {'+
                'max-width:100% !important;'+
                'overflow:scroll !important;'+
                'box-sizing:border-box !important;'+
                'height:auto !important;'+
            '}'+
        '</style>';
        $goog.prepend(stylesHtml);
    } else {
        setTimeout(changeGoogleStyles, 50);
    }
}
changeGoogleStyles();

Utilizing the Google Translate widget involves embedding an iframe with content from a different domain, specifically files hosted on Google servers. Initially, attempting cross-site scripting to manipulate the iframe content proved unsuccessful for me. However, I devised an alternative approach by downloading and modifying two essential files utilized by the widget.

It is important to note that Google may make changes to its API in the future, requiring adjustments to any implemented hacks.

Requirements:
Assuming that the Google Translate widget functions correctly on your website but requires optimization for smaller screens, below is a sample of the initial code structure:

<div id="google_translate_element"></div>
<script type="text/javascript">
    function googleTranslateElementInit()
    {
        new google.translate.TranslateElement({pageLanguage:'de', layout: google.translate.TranslateElement.InlineLayout.SIMPLE}, 'google_translate_element');
    }
</script>
<script type="text/javascript" src="//translate.google.com/translate_a/element.js?cb=googleTranslateElementInit"></script>

If your original code differs from this example, adjustments will be necessary as per your specific setup.

Tools Utilized:
Primarily, Chrome DevTools were employed (can be adapted for other browsers).

Procedure:

1. Right-click on the page containing the Google Translate widget using Google Chrome.
2. Select Inspect. This action will reveal a window or side pane showcasing various HTML information.
3. In the top menu, navigate to the Sources tab.

4. Locate the following path within the sources tree:

   /top/translate.google.com/translate_a/element.js?cb=googleTranslateElementInit
   

5. Click on the file displayed within the tree structure to view its contents.

6. Below the code snippet of element.js, there is a button with curly brackets { }. Clicking this button helps organize the code for better clarity, aiding in subsequent steps.

7. Right-click within the code section of element.js and choose Save as… to save the file within your website's file hierarchy. For instance:

   /framework/google-translate-widget/element.js
   

8. Adjust your <script> tag to reference the local element.js file.

   <!--<script type="text/javascript" src="//translate.google.com/translate_a/element.js?cb=googleTranslateElementInit"></script>-->
       <script type="text/javascript" src="../framework/google-translate-widget/element.js?cb=googleTranslateElementInit"></script>
   

9. Confirm that your website now fetches element.js locally. Verify this change using Chrome DevTools to ensure the sourcing location under:

   /top/[your domain or IP]/framework/google-translate-widget/element.js?cb=googleTranslateElementInit
   

10. The next step involves acquiring another necessary file from Google servers. Navigate through the sources tree to find:

    /top/translate.googleapis.com/translate_static/css/translateelement.css
    

    Upon locating the file, download it and click the curly brackets { }. Save the file within your website's directory structure like so:

    /framework/google-translate-widget/translateelement.css
    

11. Edit line 66 in your website files' element.js:

    //c._ps = b + '/translate_static/css/translateelement.css';
    c._ps = '/framework/google-translate-widget/translateelement.css';
    

    This modification ensures that your website loads translateelement.css locally. Verify this change accordingly.

12. Open your local translateeleent.css and append the provided styles at the end of the file. These styles aim to enhance visibility on small screens and are sourced from:

    /* Make all languages visible on small screens. */
    .goog-te-menu2 {
        width: 300px!important;
        height: 300px!important;
        overflow: auto!important;
    }
    .goog-te-menu2 table,
    .goog-te-menu2 table tbody,
    .goog-te-menu2 table tbody tr {
        width: 100%!important;
        height: 100%!important;
    }
    .goog-te-menu2 table tbody tr td {
        width: 100%!important;
        display: block!important;
    }
    .goog-te-menu2 table tbody tr td .goog-te-menu2-colpad {
        visibility: none!important;
    }
    

    Credit goes to the source from where these styles were borrowed: Google translate widget mobile overflow

13. While the styling modifications may yield positive results regarding geometry, a critical component - the text displayed by the widget (“Select Language”, etc.) - becomes fixed to one language post-hack. To address this issue and cater to non-English speaking users, amendments must be made to ensure multilingual support. Check element.js for language settings around lines 51 and 69:

    c._cl = 'fr';
    

    _loadJs(b + '/translate_static/js/element/main_fr.js');
    

    Incorporate the following correction on line 51:

    c._cl = 'auto'; //'fr';
    

14. Line 61 necessitates a more intricate adjustment due to the absence of an 'auto' value. Referencing the main.js file available on Google servers provides English as a fallback option, albeit we prefer the user's language. Examine the following location for clues:

    /top/translate.googleapis.com/translate_a/l?client=…
    

    This file contains information about supported source and target languages for translation. Validate if the browser's language setting corresponds to any target languages using the JavaScript constant navigator.language.

15. Modify line 69 in element.js with the added logic to determine and render the Google Translate widget in the appropriate language based on the user's browser preferences:

    // determine browser language to display Google Translate widget in that language
    var nl = navigator.language;
    var tl = ["af","sq","am","ar","hy","az","eu","bn","my","bs","bg","ceb","ny",
              "zh-TW","zh-CN","da","de","en","eo","et","tl","fi","fr","fy","gl",
              "ka","el","gu","ht","ha","haw","iw","hi","hmn","ig","id","ga","is",
              "it","ja","jw","yi","kn","kk","ca","km","rw","ky","ko","co","hr",
              "ku","lo","la","lv","lt","lb","mg","ml","ms","mt","mi","mr","mk",
              "mn","ne","nl","no","or","ps","fa","pl","pt","pa","ro","ru","sm",
              "gd","sv","sr","st","sn","sd","si","sk","sl","so","es","sw","su",
              "tg","ta","tt","te","th","cs","tr","tk","ug","uk","hu","ur","uz",
              "vi","cy","be","xh","yo","zu"];
    var gl = "";
    if( tl.includes( nl )) gl = '_'+nl;
    else
    {
        nl = nl.substring(0, 3);
        if( tl.includes( nl)) gl = '_'+nl;
        else
        {
            nl = nl.substring(0, 2);
            if( tl.includes( nl)) gl = '_'+nl;
            else gl = '';
        }
    }
    _loadJs(b + '/translate_static/js/element/main'+gl+'.js');
    //_loadJs(b + '/translate_static/js/element/main_fr.js');
    

This comprehensive process should provide the desired outcomes.

Consider utilizing the following CSS snippet:

.custom-container, .custom-item { width: 150px !important;}

You have the flexibility to adjust the dropdown width by modifying the value of '150px'.

This solution should be effective. Feel free to reach out if any issues persist and I will gladly reassess.

In an effort to enhance the usability of Google Translate's iframe content, I have made adjustments to allow for scrolling functionality after the initial data has loaded.

function customizeGoogleTranslate() {
    new google.translate.TranslateElement({
        pageLanguage: 'en',
        layout: google.translate.TranslateElement.InlineLayout.SIMPLE,
        autoDisplay: false
    }, 'google_translate_element');

    setTimeout(function() {
        var translateIframe = document.querySelector('iframe.skiptranslate');
        if (translateIframe) {
            translateIframe.style.maxWidth = '100%';
            translateIframe.contentWindow.document.body.style.overflow = 'scroll';
        }
    }, 1000);
}

We encountered an issue where the content within the 'skiptranslate' iframe of Google Translate was not scrollable.

Below is a simpler solution compared to others provided:

If you already have jQuery installed on your page, you can enhance the googleTranslateElementInit function that initializes the iframe with these additional steps:

<script type="text/javascript">
        function googleTranslateElementInit() {
             new google.translate.TranslateElement({
                  pageLanguage: 'en',
                  layout: google.translate.TranslateElement.InlineLayout.SIMPLE
             }, 'google_translate_element');
             
             // Set the maximum width of the iframe using jQuery
             jQuery('iframe.skiptranslate').css('max-width', '300px');
                    
             // Add an aria element for accessibility
             jQuery('.skiptranslate').attr('aria-label', 'Google Translate');
    
             // Modify the 'scroll' attribute of jQuery('iframe.skiptranslate') > #document.body
             jQuery('iframe.skiptranslate').on('load', function () {
                   jQuery(this).contents().find('body').attr('scroll', 'yes');
                   jQuery(this).contents().find('body').attr('style', 'margin:0px;overflow:auto');
             });
        }
</script>

No jQuery? No problem! Check out this alternative using plain JavaScript:

<script type="text/javascript">
    function googleTranslateElementInit() {
        new google.translate.TranslateElement({
        pageLanguage: 'en',
        layout: google.translate.TranslateElement.InlineLayout.SIMPLE
        }, 'google_translate_element');


        // Add an aria-label for accessibility using plain JavaScript
        document.querySelectorAll('.skiptranslate').forEach(function (element) {
        element.setAttribute('aria-label', 'Google Translate');
        });

        // Adjust the 'scroll' attribute of iframes with class 'skiptranslate'
        document.querySelectorAll('iframe.skiptranslate').forEach(function (iframe) {
        iframe.addEventListener('load', function () {
        var body = this.contentWindow.document.body;
        body.setAttribute('scroll', 'yes');
        body.style.margin = '0px';
        body.style.overflow = 'auto';
        });
        // Set the maximum width of the iframe
        iframe.style.maxWidth = '300px';
    });
  
  }
</script>