I have been responsible for hiring employees and I know there are different degrees of background checking. If I hire someone to stay in my home with my dogs, I would like to know that the persons background check has been thorough. This would include criminal background checks from national data bases.
I think contacting Rover HQ at (888) 453 - 7889 would be the best way to get this particular question answered, as many of us are sitters and while some of us have had our background checks done - I can't say for sure that it's "x' or "y" or even "z" background check.
Please remember that there is also the option to have someone care for your dog in their home, if trust is an issue for you!
I think Daniela's answer is spot on, but would like to add that, regardless of background check, you should make sure to schedule a Meet and Greet before booking your service. Your gut is an excellent indicator.
http://Rover.com does perform a two-part background check on all sitters attached to this company, so rest assured, the sitter you find on http://Rover.com does not have criminal history. However, I agree with Carmen. Meet and Greets are essential prior to selecting your sitter. Based on your reaction and your pet's, it will help you decide if that sitter is a right fit for your fur-family.
Once you have found that perfect dog sitter, stick with him/her. You and your pet will create bonds with that sitter, making each visit more relaxing for your pet, and will give you peace of mind.