“Is a minor child held responsible for unauthorized charges made on a debit card?”
Under the Electronic Funds Transfer Act, the liability for the holder of an account is limited to $50 if the bank is notified within 48 hours of the misappropriation of a card, and $500 if the bank is notified within 60 days. If you wait more than 60 days to notify the bank, then you could be on the hook or the entire amount.
If it is the minor misappropriating funds, then laws governing theft in the State will apply. Chances are pretty good that there will be some liability to the victims of the crime even if the thief is a minor.