What should I do about nonmetallic sheathed cable that was wet?
The water alone was NOT what tripped the breaker. Water alone does not cause shorts and arcs. You have another issue somewhere in that cable run. Check every connection, box, splice, etc. 

And yes, you could be doing more damage to the wire or breaker by repeatedly trying the breaker and tripping it so many times. 

On another note, if you only have 12/3 going to a typical electric dryer (in the US I assume) that is NOT the correct cable. An electric dryer requires 10/3 on a 30A breaker.