Every website owner can and in my opinion should encourage people to update their out-of-date browsers. All they have to do, is adding a little warning to their site in the moment a legacy browser is detected.
I'm sure a few months ahead this won't be an issue at all. Browsers, like today Chrome, will automatically update and therefore forcing updates.
But till then the website owners are in charge to enlighten their users.
Just add the unobtrusive script from browser-update.org to your pages (direct before the </body> tag):
Then users with an old browser would see a non-intrusive warning at the top of the page: