You can post messages to the screen, encouraging visitors to upgrade or switch browsers, but this technique is far less necessary today than it was just a few short years ago.
Today, most modern browsers play well with websites that are implemented in compliance with applicable design and development standards. And most of these browsers self update to a great degree, unless specifically blocked from doing so (as is the case in some large, tech unsavvy organizations). Browsers that are blocked from updates can not be "fixed" simply because your website is asking them to update, so your messages will likely do nothing more than alienate some of your visitors.
The better strategy is to design the website to gracefully dumb down when it encounters browsers that cannot perform certain functions.