Whoever owns the credit card accounts is responsible for payment. If your dad is the sole owner of the accounts, then he alone is responsible for payment. If any of the accounts are joint ownership, then your dad and the other person named as a joint owner would be equally responsible for payment.
My experience tells me things are not always as simple as they should be. So I suggest you ask your dad to let you get copies of his credit reports for him for free at AnnualCreditReport.com. Then you can confirm whose name(s) is on the accounts.
In general terms, when your dad passes away, it will be up to the estate's executors to notify his creditors and the credit bureaus. This will prevent possible identity fraud and be a courtesy to his creditors. His creditors would have to file a claim against the estate to collect any outstanding debts in his name. State laws determine how claims must be filed. It is hoped your father has a will. If he doesn't then that's another topic for you two to have a conversation about. When notifying creditors of a death, make sure to include a copy of the death certificate and the estate's contact information. This will prevent the creditors from contacting you or your other siblings looking for payment.
If there is no money in the estate, don't be surprised if they ask you for a payment even though you are not responsible. Getting loved ones to make a payment during a time of distress is very common, especially if the debt is in a collection status at the time.
That's for the future (and I hope well into the future); now a quick word about the present. It sounds as if your dad may be using credit to extend his income. With his cards maxed out, he has run out of credit to use. This could be the precursor to a default and collection action if he isn't careful. You might want to have a conversation with him about his finances, and determine how he is doing.
Some options he might consider include a consolidation loan, a balance transfer to another card or adjusting his budget to see if he can lower his expenses and keep up with his payments while living within his income. I wouldn't suggest he use his home's equity or a reverse mortgage unless they are part of an overall financial plan. Those loans are secured by assets that may require property to be sold at his death.
Once you have learned more about your dad's finances, then you will know what makes the most sense. As long as your dad's credit is still good, he may qualify for a loan to consolidate all the balances. However, he will need to have enough income to make the minimum payment due once all the debt is transferred.
Ask the adviserTo ask a question of the Debt Adviser, go to the "Ask the Experts" page and select "Debt" as the topic. Read more Debt Adviser columns and more stories about debt management.