Will my New York auto accident case be affected because I gave a recorded statement to the other driver's insurance company?

That could very well happen and it may happen even if the statement was given honestly and innocently, because anything you say can be used against you, and it can be misconstrued against you as well. You're not going to be in a position to figure out how a particular answer is going to affect your case. The prudent thing to do is to not give any statement to any insurance company with the other side involved in the accident. It can only hurt you. It really can't help you.