If you LIVE in New Jersey, you are supposed to file a New Jersey tax return no matter whether you own property in NJ or just rent your residence. But...you work in New York. Is your employer withholding New York tax? You are supposed to file state returns for each of those states. But your New York return would need to be a non-resident return. When you live in one state and work in a different state, you need to prepare your NON-resident state return first, then prepare the state return for the state you actually live in. That way, New Jersey knows you paid tax to NY, and NY knows you paid tax to NJ, and you will not be "double-taxed."
https://ttlc.intuit.com/questions/2895920-why-would-i-have-to-file-a-nonresident-state-return
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.