Kait--Well, it seems that weather right now is unseasonably warm (even with this "cold front" we've been having, warmer than usual). March is usually okay as far as weather goes-we go into NYC every March for ds's birthday. If you're going as a tourist (esp with kids, believe it or not), it's not so bad. Go into ESPNZone, walk 50 ft, go into Toys-R-Us and warm up (and that place is huge), walk 50 ft, go into a tourist shop, walk 100ft into a star bucks and get a cocoa, walk across the street into Bubba Gump's and warm up...well, you get the idea.
But like anyother place in this neck of the woods, the better months would be April, May, June, (July and August as long as it's not a super hot day--super hot for us being in 90s), Sept....
And to the OP (or anyone else reading)--Please try and see the "Bodies Exhibition" I think it's about $25 per person and it's AMAZING! Real human bodies in diff positions, preserved with silicone and polymers. Awesome! (and it's at the South Street Seaport, if you end up staying there) It's leaving NY at the end of Feb.
What you might think about doing is staying on Long Island hotel and take the train into NYC each day. You could pay $150 a night and get a LOT more than you'd get for that price in Manhattan (an indoor pool and much bigger room--as opposed to two beds crammed into a closet, which is what you'll get in NYC). There are lots of things to do in Nassau County (the county bordering NYC)--Long Island Children's Museum, Cradle of Aviation Museum, huge mall (Roosevelt Field), historic estates and farms within 15 minutes of the hotel and train station. The LIRR-long island rail road-would be about a 30 min train ride into Penn Station-which is in the heart of NYC--walking distance from Empire State Building, Times Square (incl. broadway), and a major hub for all subways if you want to go to the seaport.
The American Museum of Natural History is very easily accessed from the subway system and is a great place to visit-with or without kids. It can be free if you're broke--though I don't suggest taking advantage of their 'donations only' admission. It also borders on Central Park, and the observatory offers some BEAUTIFUL views.
Feel free to PM me. Either way--enjoy your visit.