Why does my basement floor paint come off when it's wet?
First it sounds like you used a latex paint which for a basement floor is not a good solution.  You need to use the messier, oil based paint.  This will soak into the concrete a bit and are a lot more durable.  This paint will continue to have problems when wet - especially with hot water or when scraped by something sharp or heavy.

Also at the 3 week period I doubt your paint has fully cured in a basement environment.  Once it cures it will hold better but probably not up to your expectations.