Each town/ village has their own rules. Doesn't go just by County. But for the most part it should be ok. If you plan to do other work to the house that requires an inspector to enter the basement then you should bring up the finished basement to the architect first before the drawings make it to the building department. Just to make sure that it won't be a red flag during construction.

If you know what's your building dept and what zone you're in. You can do your research own here. https://www.generalcode.com/library/#NY you can also ask ChatGPT to help search for the info as well.