You claim the "other state credit" on your home state's tax return. In your case, NY will credit you for the taxes you pay to NJ on the dollars that are taxed by both states. This prevents double taxation, which federal law prohibits.
This is why it is key in TurboTax to complete the non-resident state tax return before the home state return. That sequence allows the program to correctly calculate and apply the credit.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
