Yes. Speak with your employer.
Tax Champ @Rick19744 has excellent points. New York is very aggressive about chasing people who work in other states who do not pay New York income tax.
Your employer's withholding is not a 100% guarantee that they are right. However they do know the law and your situation, especially if they've been doing this across other states with multiple employees.
The safest route is to report your income to New York. Since there was no NY withholding, you would owe a lot. You can claim a credit on your Kentucky return for tax paid to NY and get a KY refund. The NY tax rate is usually higher so you'd pay more total tax and you may not be required to pay NY tax.
Once you start that route, you'd have to continue. Doing it one year and stopping would attract attention.
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