It depends. For these situations, you must file two state tax returns: a NY nonresident return, and a NJ resident return. Both states tax you on the income. NJ will give you a credit for the tax you pay to NY on the income you are earning there. You will prepare the NY return fist to allow TurboTax to calculate that credit accurately. Here is an FAQ which provides more information on this: https://ttlc.intuit.com/replies/3302052
On the NJ return, you will have a couple of screens to make sure you have correct entries to ensure that the income is not being counted twice. First, you will be asked about your income. You will follow the screenshots below to exclude the line on your W-2 regarding your NY income. Later, you will see the screen about the tax credit to NY. You will want to make sure that the double-taxed income has been correctly credited. Please note the following 4 screenshots to assist you:
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