There is already a little back and forth from the locals (of which I'm one, from NJ actually) as to which airport to fly in to, and I'd suggest you should take that as a hint that you don't have really convenient options here in terms of GA airports to fly to in order to easily access NYC on a one day trip for sight-seeing in Manhattan -- not to say it can't be done if you're really ambitious and ready for a long day.
So I'd suggest you think carefully about your planning, logistics for getting around, and the time it will take you to do it all. Maybe your best bet, considering you're starting out with a 40 minute flight from Binghamton, is to just fly in and do the awesome Hudson River Corridor flight for spectacular sight-seeing (Statue of Liberty, Ground Zero, New Liberty Tower, Manhattan skyline, George Washington Bridge, Verrazzano Bridge, etc.) and call it a day and return to Binghamton. If you do, make sure you understand the airspace, procedures and self-announcements you are to make in the corridor. See here:
http://www.faasafety.gov/files/gslac/courses/content/79/776/kneeboard.pdf
If you want an airport with a restaurant to stop at on the way back, you could go to Blairstown (1N7) in Northern, NJ, which is a nice country airport (gliders, etc) right on your route with a decent cafe/restaurant (
http://www.blairstownairport.com/default.htm).
I haven't journeyed from either Linden or Republic Airports into Manhattan, but you can see that they are really not that close. Google maps shows Republic at about 40 miles drive by car to downtown Manhattan, while it shows Linden to be about 26 miles. It's already been said that the train ride from Linden to Penn Station in NYC is about 40 minutes, and the trip from Farmingdale is closer to an hour. Assuming you don't rent a car, that leaves you to get from the airport to the closest train station by cab or otherwise, wait for the next train to arrive, then take it to NYC and start traveling around to your final destination within the city from there. That's going to take you some time, and obviously you're going to have to do it in reverse when you're finished, then get back in your plane for your 40 minute ride back to Binghamton.
If you think you can overcome the obstacles, I'd vote Linden, since it's (a) closer on your way from Binghamton, and (b) closer to NYC than Republic, and (c) only about 1.5 miles to the NJ Transit train station (but you will need a cab for that). However, to get in to Linden, you will be flying at low level under Class B over some densely populated areas with basically no landing areas, so I agree with Vlad it's intimidating. Here's the NJ Transit train schedule for Linden to NYC:
http://www.njtransit.com/pdf/rail/R0070.pdf
However, my vote, if you decide to go for this marathon, would be to not take the train from Linden all the way to Penn Station at 34th St. in central Manhattan, but rather to get off at Penn Station in Newark (that's right, Newark and Manhattan each have stations confusingly named Penn Station, but one is in Newark and the other in NYC) and transfer onto the "PATH" train to the World Trade Center station in lower Manhattan, where you will get off right at the World Trade Center and can check out the 9/11 memorial, etc. You can see the PATH schedule on the NJ Transit schedule linked above. That might be a trip into Manhattan you could accomplish in one day coming from Binghamton with an early start.
My other vote, if you had the time, for sight-seeing in NYC with kids would be to go to Battery Park, at the lower tip of Manhattan, with a beautiful view of NY harbor and the Statute of Liberty, with a little museum of the old fort, and then take the ferry ride tour to the Statue of Liberty and Ellis Island. What a great trip that is, for you and the kids. Here's the info here:
http://www.statuecruises.com/ The tour tickets are surprisingly cheap, compared to everything else in NYC, where fleecing tourists is an art. To get there, you would also take the PATH to the World Trade Center station, and you could then actually walk to Battery Park, going right past the NY Stock Exchange. It would take an early start and serious ambition, though, to pull that whole trip off in a day coming from and returning to Binghamton, so I'm not sure I'd advise it. You as the pilot would surely be pooped at the end of such a day. You'd never forget it though.
Good luck!