If you live in NY and work in NJ (or vice versa), you must file a tax return in both states. In TurboTax, you must prepare your non-resident state return first, before you do your home state return. If you prepare the returns in that order, you'll receive a credit on your home state return for the taxes paid to the non-resident state, and you won't be double-taxed.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.