In my opinion, you should hire an attorney to handle this for you. All states share information regarding traffic violations, so if you plead guilty by paying the fine, that conviction will be reported to your home state of Connecticut.
Why do you think you could successfully fight it? Are you counting on their not being willing to bring the trooper in fro a hearing? Good luck with that. Better talk to some New York lawyers and be sure you know the drill.
BTW Connecticut has use a "points" system since Ronald Reagan was President.
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Out of state points/violations transferring depend on whether the states have reciprocity (share fines and violations without asking) or if they routinely abstract information from other states (request reports, etc). I would pay the fine, but I don't know anything more about your situation than you've posted. States other than Connecticut can charge you to fight the ticket. The court costs can be more than the fine.
You might want to order a copy of your driving record from NY at the end of the year. Sometimes if an out of state driver pays the fine they just disappear...