Hittade detta på the interwebs.. kanske värt att testa..
One trick that works on most browsers I've tried it with is to make the person go to a part of th site for which they are not authorized.
For instance, make a directory "/logout" and make the .htaccess there use a different file of usernames and passwords. If your user goes there, they will get asked for a username/password, but when they try theirs, they will get denied. This makes the browser forget the password. Your auth denied page for this directory should be something like "You have been logged out".
It's pretty crude, and somewhat confusing, but there aren't many other alternatives. That's one reason not a lot of people use basic auth such as with .htaccess.