Note that I am not licensed to practice in your state.
His estate is responsible for debts. Any money that he left behind will go to satisfying debts. The debts will not be passed onto family members or his children.
The simplistic answer is that the card-holder's estate is obligated, and nobody who is a beneficiary will "inherit" the debt, it just serves to reduce or extinguish estate assets.
There are exceptions, however, and this depends largely upon two things: The marital property system of the state of decedent's residence, and the characterization of the debt. In a community property state such as California, for example, even though the card is in the decedent's name alone, if he was married at the time the debt was incurred and at time of death, and the debt on the card was for community necessities, rather than for something clearly not a community expenditure, then an apportionment can take place. The community property of the spouses is responsible for community debt, no matter which spouse incurred it. Each spouse is the agent of the marital partnership, insofar as incurring debts for necessities of life, i.e., for the benefit of the community of the two of them. Generally banks won't want to dig into the nature of the debt and chase a surviving spouse for her share of it, but if it is a significant amount and she appears worth chasing, sometimes this will happen. Her defense will be to refute the notion that the balance represents expenditures for the community.
