What sort of background checks do they do for foster parents?

Do they just do criminal or can they look at your medical record as well?
Answers:
Different localities have different levels of scrutiny - and it's not just a variance by state. For example, the City of New York's Agency for Child Services handles its checks differently than does the similar agency in Syracuse.
But, depending upon the agency, they can look at everything from criminal to financial to medical records. And they should!

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Depends on the state. Check out your state's web site for foster parent qualifications. Usually it is more than just a criminal background check. They usually do a character background check as well, i.e. family members, neighbors, your own child(ren)s teachers, etc. and credit checks as well.