Who is responsible for an adult child's debt after their death?

Provided the "child" is really an adult, check your state laws but generally 18 or 21, their estate is responsible for paying their debts after death. If they owed more than the total owed to all creditors at death, then the creditors are paid proportionately out of the estates assets. Any unpaid balance once the estate is empty the creditor has to write-off. No one else is responsible for these debts. If, on the other hand, a parent or family member co-signed for them, then the creditor can hold the co-signer responsible, but only for that account.