1st question was your income greater than $6k if not, no non-resident return in CT. The rules for CT (then see comments below on if CT taxes are less than place work performed) <a rel="nofollow" target="_blank" href="https://portal.ct.gov/DRS/Nexus/Connecticut-Source-Income-of-a-Nonresident---Nexus">https://portal.ct.gov/DRS/Nexus/Connecticut-Source-Income-of-a-Nonresident---Nexus</a> "Attributable to compensation for services performed in Connecticut or income from a business, trade, profession, or occupation carried on in Connecticut (including income derived directly or indirectly by athletes, entertainers, or performing artists from closed-circuit and cable television transmissions of irregularly scheduled events if such transmissions are received or exhibited within Connecticut);"
I can add further that you may also have to pay non-resident Mass (I am assuming MA is Mass). Mass requires a non-resident return for income over $8k. "Compensation for personal services performed in Massachusetts, regardless of where paid. This includes but is not limited to wages, salaries, tips, bonuses, commissions, fees, and other compensation related to activities carried on in Massachusetts, regardless of where the compensation is paid. In the case of compensation for personal services, report all Massachusetts source income even though you didn't receive the entire amount. For example, include employer withholdings for federal income taxes, FICA contributions, medical insurance plans, or other similar withholding deductions in Massachusetts source income." <a rel="nofollow" target="_blank" href="https://www.mass.gov/service-details/learn-about-massachusetts-gross-income-for-nonresidents">https://www.mass.gov/service-details/learn-about-massachusetts-gross-income-for-nonresidents</a>.
Non-Massachusetts employers
If you're a non-Massachusetts employer who conducts business or maintains an office in Massachusetts, you must withhold the amount determined for Massachusetts income tax purposes, minus any amount deducted and withheld for your own state. If there's no state withholding in your business's home state, you must withhold and send to us the full amount determined for Massachusetts purposes. Whether the employee works on the road, at home or in an office maintained by you, the employer, does not affect your withholding obligation. <a rel="nofollow" target="_blank" href="https://www.mass.gov/guides/withholding-taxes-on-wages">https://www.mass.gov/guides/withholding-taxes-on-wages</a>
You would complete the non resident returns first and only then complete the NY return. You would get a credit on the NY return for double taxed income. During the personal section of the federal return, only answer yes that you worked in another state, IF you met the criteria needed to file in them.
You are not required to file a CT return, if all your work was performed in MA, unless you had CT taxes withheld, in which case you would file to have them returned to you.
@Zbucklyo please read my comment above as to what CT sources and MA sources as income.
Mass must withhold income tax for any business performed within Mass, they are getting more like NY.If you're a non-Massachusetts employer who conducts business or maintains an office in Massachusetts, you must withhold the amount determined for Massachusetts income tax purposes, minus any amount deducted and withheld for your own state. If there's no state withholding in your business's home state, you must withhold and send to us the full amount determined for Massachusetts purposes. Whether the employee works on the road, at home or in an office maintained by you, the employer, does not affect your withholding obligation. <a rel="nofollow" target="_blank" href="https://www.mass.gov/guides/withholding-taxes-on-wages">https://www.mass.gov/guides/withholding-taxes-on-wages</a>
CT; The employer will withhold tax for the state in which the individual works and will submit the tax to that state. If the employer is also registered to withhold Connecticut income tax, Connecticut income tax must be withheld by the employer, but only to the extent that Connecticut income tax withholding exceeds the amount required to be withheld for the state in which the services are performed. For further information, employers should consult the Circular CT, Connecticut Employer's Tax Guide (so if CT tax rate is higher only then CT taxes must be withheld. Connecticut has a progressive income tax structure, with rates ranging from 3 percent to as high as 6.99 percent, while Massachusetts has a flat 5.15 percent income tax rate so depending on income will depend if CT tax should be paid ).
Determining How Much to Withhold From Nonresident Employees Who Work Partly Within and Partly Outside Connecticut . . . page 12
<a rel="nofollow" target="_blank" href="https://portal.ct.gov/-/media/DRS/Publications/pubsip/2018/IP-2018(1).pdf?la=en">https://portal.ct.gov/-/media/DRS/Publications/pubsip/2018/IP-2018(1).pdf?la=en</a>