Are the children of a deceased parent responsible for their debt in the state of New Jersey?

Yes and no. The estate of the deceased is responsible for payment of any debts of the departed, so if you had been willed any form of an asset then it could be liquidated or levied for the repayment of a debt owned to the estate. The only time children or a surviving spouse are liable is if they put themselves up as a guarantor, including co-signing on a car, apartment, credit card, and/or medical care.