Karen,
I understand your apprehension. But think about it, many of us have left our children with a "sitter" when only meeting them once, so why not a pick up person? I get that you are letting someone drive your daughter around, not just sit in the house, but still...
I agree, get a copy of her license and insurance card. Ask her if she's ever been in accidents/had tickets. Check her references and perhaps to a background check. Also, be clear, maybe in writing, what your expectations and her pay rate will be. As in, is it per pick up? Is she allowed to have friends in the car when she's working for you? Make sure your daughter sits in the back seat. Make sure she gets out of the car to get your daughter, so she isn't wandering aimlessly looking for the woman/girl.
Obviously you get the idea.
I think the expectations in writing is as important as the background check. Especially because she is going to be driving and unsupervised with your child. This may help you and your DH feel more comfortable too.
I hope this helps!