How can I make the App Store remember my password?
You can't change this behavior.  Although, when you download a free app, you have to enter your password; if you download another free app immediately after, you won't have to re-enter your pass.  That's as close as you're going to get to a remember password feature on iOS 5.

This is something that was intentionally in place and now it's intentionally changed on iOS 6 to reduce the need to re-enter your password just for updates.