Your husband having criminal record, not working and being on public assistance won't jeopardize your chances to be a legal resident.
Since he is not working, then of course you would need a co-sponsor to overcome the presumption of becoming a public charge.
Him having a criminal record doesn't mean you have a criminal record. INS only cares about your criminal record, and not anyone else because you are the beneficiary here for an immigration benefit. That's why they run a FBI check on the beneficiary. Thus, him having a criminal record means nothing in your case.
Since he is on a public assistance then officer might like to know for how long because many of public assistance are determined based upon household income. Yet still adjudication officer won't deny your application for green card if you would have a co-sponsor. But if you ever been on a public assistance then officer will deny your application.
Don't worry about all this. And, don't listen to others if they tell you that it will question the bonafide of your marriage or anything else. Immigration officers face this kinds of cases on daily basis. Thus, they are aware of all this situations. It has nothing to do with the fake marriage or digging up more in a case.
Good Luck.