YES, every person hired gets a background check. Once you complete the check it comes back within a few hours to the store. It's a quick process which is why you can be hired on potentially the same day or the day after your interview.
I think they might perform background checks
Answered - Sales Associate (Former Employee) - New York, NY
They say they do but they hired me the day after my interview
Answered - Sales associate (Current Employee) - Pembroke Pines, FL
DSW does background checks as well as lengthy (but worthwhile) personality tests.
Answered - Warehouse Associate (Former Employee) - Alpharetta, GA
Yes I believe they do
Answered - Picker/Packer (Former Employee) - Columbus, OH
Yes, your offer is contingent on a background check.