K.P.
NYC is expensive. Period. That is a LONG drive with three children to be in the City for essentially 36 hours. NYC as a tourist "is" Broadway and museums/shopping. You could "splurge" and spent an entire day b/w Central Park and the Museum of Natural History. If you do the entire thing, it takes a full day. It's pricey, but would be fun for your kids. The Bronx Zoo is another full-day option.
**NYC in July is HOT!** I know that I'm being negative here, but living in the Hudson Valley and working in Westchester I can assure you that there is NOTHING inexpensive surrounding the City (including gas). This is going to be an expensive trip and an incredibly long drive to go somewhere that is not "kid friendly on a budget".
Just a thought.... why not bring the family to the Hudson Valley for the weekend? It's much less expensive and very family-friendly. There are so many things to do here and you could stay in a quaint town like Rhinebeck or Staatsburg for much much less!