That income is taxable by both states. NJ taxes its residents on all their income, regardless of where it's earned. NY can tax non-residents on income earned from work performed in NY. So you must file both a non-resident NY return and a resident NJ return. NJ will give you a credit for the tax you pay to NY, so in effect you won't be double-taxed.
In TurboTax, complete the non-resident NY return before you do your home state NJ return, so that the program can calculate and apply the credit.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
