working remotely in CT while a resident in NY

My son is a NYC resident working for a firm based in NYC but spent about 3 months working remotely from our home in CT.  He billed hours from CT so there is a CT state W-2 with CT withholding.  New York State requires that tax is paid on the entire earnings even though he was in CT at the time.  I read that CT will credit the taxes withheld in CT but I don't know how to go about filing the CT state taxes, and if there are additional steps to file.    Also, as a result, NYS did not withhold enough taxes so there is a significant balance due.  Thanks in advance!!