Most users surfing on archaic browsers have faced this situation not once, but many times. So chances are they're fully aware of what's going on. Whether they don't change it for being stubborn, lack of knowledge or being forced to it (some enterprise level software still uses IE8), is not your problem.
However, it's your problem to serve the best possible experience on YOUR site.
So, first, try to accommodate your site to the broadest possible array of browsers, and test your site. After this, assuming there are issues with any specific browser (which is very understandable, you can't go back to stone ages just because someone didn't upgrade), just let the user know BEFORE they embark into a journey they won't be able to complete.
Additionally, it would be a good idea if your site has good mobile support since some of your users will be able to use your site on mobile/tablets