If you're a resident of NJ, then ALL your income is taxable by NJ, regardless of where you earn it. In addition, any income you earn from work you actually (physically) perform in NY is taxable by NY.
You must file both a non-resident NY tax return and a home state NJ tax return. NJ will give you a credit for the taxes you pay to NY, so in effect you won't be double-taxed.
As @xmasbaby0 said, with TurboTax, you must complete your non-resident tax return before you do your home state tax return, so that the program can correctly calculate and apply the credit.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
