This kind of thing is not really reasonable.
You'd have to have a way to always know the latest version of each browser, and since no standardized way exists, you'd have to resort to screen-scraping the browser vendors sites to find it. If the pages you are scraping change, you'd have to rewrite your scraper.
On top of that, you'd have to check that a later version of the browser is available on their OS. If they're on XP using IE8, they can't upgrade. If they're on old ubuntu using old chromium, they can't upgrade, etc..
The usual way to handle this is to officially support minimum versions of a few browsers, and possibly notify the user if they're not using a supported browser or version.