Your earnings from work actually performed in NY are taxable by NY. If CT is your state of residence, ALL your income is taxable by CT, including the NY income.
In that situation, you would file a non-resident NY tax return in addition to your home state CT tax return.
You'll be able to take a credit on your CT return for taxes paid to NY, so you won't be double-taxed.
In TurboTax, complete the non-resident state return first, so that the credit flows properly to the home state return.