Dana Ross

This plugin displays a message at the top of the screen for visitors running outdated versions of the most popular browsers:
<ul>
<li>Internet Explorer 6 SP2 or less</li>
<li>FireFox 2 or less</li>
<li>Safari 2 or less</li>
</ul>
It is designed to be unobtrusive. The message disappears after a few seconds, and it only displays the first time the visitor comes to your WordPress blog/site.
The warning reads: It appears that you are using an outdated version of your browser. Please upgrade your browser for better sercurity and an improved experience.
The message also includes a link where the newest version of the browser can be downloaded.

Dave’s Outdated Browser Warning

MacSteini

Many users still browse with outdated browsers, often unaware of the risks. Upgrading ensures better security and reliability. This plugin displays a subtle notification prompting visitors to update their browser. Activate the plugin, and it works seamlessly.
Visit <a href="https://browserupdate.org/" rel="nofollow ugc">browserupdate.org</a> for more details.
Want to help translate this plugin? Visit the <a href="https://translate.wordpress.org/projects/wp-plugins/wp-browser-update" rel="nofollow ugc">WordPress Translation Project</a>.
<h3>Important Notice</h3>
Breaking Changes in Version 5.0 
– Requires PHP 7.4 or newer. 
– Ensure your hosting is updated to PHP 7.4 before upgrading to version 5.0 or newer. 
– Servers running older PHP versions are no longer supported. 
 – If your server is running an earlier PHP version, please download <a href="https://downloads.wordpress.org/plugin/wp-browser-update.4.8.1.zip" title="Download WP BrowserUpdate from WordPress.org" rel="nofollow ugc">version 4.8.1</a>.

WP BrowserUpdate

Used for a client who had this exact request. Worked well.

Works as expected

This plugin notifies website visitors to update their outdated browser in a non-intrusive way.

Dave’s Outdated Browser Warning vs WP BrowserUpdate