Using Apache, it is quite simple to set up a page that uses basic access authentication to prompt a user for a name/password and use those credentials in some way to grant access to that user.
Is this secure, assuming the connection between the client and server is secure?
The worry about basic auth is that the credentials are sent as cleartext and are vulnerable to packet sniffing, if that connection is secured using TLS/SSL then it is as secure as other methods that use encryption.
If you are generating passwords with htpasswd consider switching to htdigest.
Digest authentication is secure even over unencrypted connections and its just as easy to set up. Sure, basic authentication is ok when you are going over ssl, but why take the chance when you could just as easily use digest authentication?
As the name itself implies, 'Basic Authentication' is just basic security mechanism. Don't rely on it to provide you with worry free security.
Using SSL on top of it does makes it bit more secure but there are better mechanisms.