Any requirement to register as a DBA is a matter of state law; e.g. Florida and Michigan have different standards. In general, the question is does the customer use his own name as that of the business or does he use a name he only "assumed."
If you wish, you can require the individual to open a business account. You can also require a copy of any local, state, or federal license the law might require him to hold. However, if I am James Longstreet and the checks are made payable to "James Longstreet" there is simply no argument that I need to register under an assumed name.