Since you're an NY resident, ALL your income - including your earnings in CT - is taxable by NY.
Additionally, CT can tax you as a non-resident on your earnings from work actually (physically) performed in CT.
Therefore you'll file two state tax returns - a non-resident CT tax return, plus your home state NY resident return.
You'll be able to take a credit on your NY return for taxes you pay to CT on the income that is taxed by both states, so you won't be double-taxed.
Since you'll only be working in CT one day a week, your employer should at a minimum withhold NY taxes. If they will also withhold CT taxes for your work in CT, so much the better.
When you do your returns in TT, be sure to complete the non-resident CT return first, before you do your home state NY return, so that the credit is calculated and applied correctly.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
