Here's how it works:
Since you're a resident of NJ, NJ can tax ALL your income, regardless of where you earn it.
In addition, NY can tax you as a non-resident on any income you earn from work you physically perform within New York State. NY can also tax the income you earn by working from home, if you're working at home for your own convenience. That's why the question raised by @rjs is critical.
In any case, you'll be able to claim a credit on your NJ tax return for the taxes you pay to NY, so in effect you won't be double-taxed. The only question is how much of your income will be taxable by NY.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
