Absofreakinlutely. In MD we are fortunate to have access to the judicial records via MD's Judiciary Case Search.
Maryland Judiciary Case Search
You can search criminal records, civil actions, judgements, etc.
I don't believe DC has that resource. For our DC rentals, we'd use SafeRent's background check, which came along with the credit check. It's not as thorough as MD's public records, nor as detailed. But it beats hiring a weirdo (or renting to one.)
First Advantage SafeRent
As for privacy issues, just make sure that they sign a release consenting to a background/criminal check as a condition of employment. Most people are expecting something along these lines. Good luck.
"The first lesson of economics is scarcity: there is never enough of anything to fully satisfy all those who want it. The first lesson of politics is to disregard the first lesson of economics." Thomas Sowell