That depends. "DBA" is an imprecise term, because it can apply to a corporation, LLC, LLP, partnership or an individual (a sole proprietor).
Clearly, a bank should never accept a POD designation from a corporation, LLC, LLP or partnership. It would be useless, since those are all entities that aren't living beings, so they don't die. State laws providing for POD designations only apply to accounts owned by living human beings. Don't be confused by the fact that corporations, LLCs, LLPs and partnerships are often owned by humans. While they may be, their accounts are not owned by the owners of the entities. They are owned by the entities themselves.
But when the subject is the account of a sole proprietor (who is a living, breathing, human), most analyses I have read agree that, unless state law has a provision to prevent it, a sole proprietor can designate a POD beneficiary on his/her business account, whether the sole proprietor operates under his/her own name, or uses a DBA (doing business as) or fictitious name for the business.
If you are uncertain whether your state laws allow it for a sole proprietor, consult an attorney before accepting the designation.