Their credit is not reported on yours unless they mess up the checking account itself. If they bounce a check or start getting a lot of NSF's that will be on both of you. But their previous credit cannot affect yours.
First, I would like to explain to you the difference between having bad credit, and having a bad report on ChexSystems.
Banks and credit unions usually report people to Chexsystems regarding delinquent bank accounts.
Financial companies and lenders report people to the 3 major credit bureaus regarding delinquent accounts.
So, with that said, having a joint checking account with someone with bad credit does not affect your credit. If that person goes into delinquent status with the checking account and the bank reports him or her to Chexsystems or some other check verification company, then yes your name and social security will be reported as well. Your report will stay on Chexsystems for at least 5 years, unless you have it removed by some means.
If you cannot remove your report from Chexsystems however, you should consider applying for a second chance checking account.
http://www.2ndchancecheckingaccount.com