Parking in NYC (or outside the city)?? For a two night visit... Coming in from Boston area?

My family and I are taking a trip to New York City for the first time. We will be in NYC on June 14-16 (Tuesday-Thursday). This will actually be part of a bigger trip through the North East...We will be coming in to NYC from the Boston area. We are staying at the East Side Marriott at Lexington and 49th street. We will actually be in two vehicles and we don't intend on using the vehicles while in NYC. The parking garage for the hotel has a fee of $60 per night per vehicle (which for 2 nights for 2 vehicles, would be $240). We are from the South and have no idea about "big city" parking/driving. My question is: Is there somewhere inside the city or outside the city (along our route) that would be a safe and cheaper option than paying for the hotel parking garage? We would be willing to ride public transportation (obviously) to our hotel after leaving out vehicles (what public transportation do we need to take?). Also, we will be arriving in the evening on the 1st night and we will have a 7 year old with us (to take this into account for safety). Please only knowledgeable answers and I REALLY appreciate the help.