This is coming from someone who .i.s a babysitter herself, and these are the things parents always ask for:
-references
-proof of certification (because .I put on my flyers that .I am first aid, cpr, and heimlich certified, and .I've taken the red cross babysitting course)
-rates
and then most parents keep me talking while .I have to entertain the kid. That way they can tell if you and their child(ren) get along, and if you're capable of watching their child.
If you're hiring an adult (which .I inferred that you are) ask to see her driver's license, and definitely make a copy of it. background checks are really important; it's the same thing as when they screen people to be girl scout leaders. they all get background checked first.
And the random drop-ins are a great idea.