My husband owes back taxes. If he dies, who will be responsible for the payment. I own a condo, which he has nothing to do with. These taxes back taxes are his alone.

If the back taxes are from returns which he filed BY HIMSELF, then his estate is responsible to pay these taxes before distributing any assets to the heirs.

The only way YOU become liable for these taxes is if they were incurred on a JOINT return. Otherwise, you are okay.