Since you're a CT resident, CT can tax ALL your income, regardless of where you earn it.
NY can tax you as a non-resident on income you earn from work actually (physically) performed within New York state, and on income you earn from a NY employer as a telecommuter from CT, if you telecommute for your own, rather than your employer's, convenience (this is known as the "convenience of the employer" doctrine).
You can take a credit on your CT return for the taxes you pay to NY state on income that is taxed by both states, so you won't be double-taxed.
Because you can take that credit, your NY employer should be withholding NY taxes. Most, if not all, of your actual tax bill will be owed to NY.
So you will be filing two state tax returns: a non-resident NY return, AND your regular home state return. In TurboTax, be sure to complete your non-resident NY return first, before you work on your home state CT return. That way, the credit will flow properly.
If your NY employer requires you to telecommute from CT, the tax situation would be different. In that case, your telecommuting income (from a CT location) would be taxable only by CT.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
