If you want to use a MySQL database to authenticate people with Basic Authentication on an Apache web server (as opposed to using .htaccess files or whatever), don't bother googling for a module name "mod_auth_mysql". You might think that's the right thing to do, but it isn't. There are a bunch of modules that were called "mod_auth_mysql". Seems like everyone who wrote an Apache module wrote one called mod_auth_mysql at one point. There are a lot of dead projects out there.
Instead, just go here and get the one by Vivek Khera. It works with Apache 1.3.x and 2.x. It works as a DSO. It actually compiles, which is nice. And it's brutally simple to use.