Is the custodial parent responsible for car insurance in nj?

I am not sure what you are asking. If a person has a car they need to have car insurance. Unless there is some court order or language in a support agreement that requires the other parent to maintain car insurance or to share in that cost, I believe it is the custodial parents responsibility. Health insurance is different, and often the non-custodial parent is required to pay at least a share of the insurance, especially if that parent is employed. I know of no such rule for car insurance.