The portion of your income that you earn by physically working in CT is taxable by CT, as well as by your home state of NY.
At year's end you'll file a non-resident CT tax return on which you report your income earned in CT, and you'll file your usual NY home state return on which you report all your income, including your income earned in CT.
You'll be able to claim a credit on your NY return for the taxes you pay to CT, so you won't be double taxed.
In TurboTax, be sure to complete the non-resident state return before you do the home state return, so that the program can correctly calculate and apply the credit.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
