I've never flown to New York, but...
Have you tried the subway route from JFK? It's a (miserable) hour or more subway ride to Manhattan.
Building a better railway to JFK seems a good idea regardless of what happens to Newark.
My nearest airport is London Heathrow, which has the cheap option of an hour-long metro ride (using that word, since most of it is above ground), but there's also the "premium" Heathrow Express train, which takes about 15 minutes, a cheaper (and slightly slower) train, and the new Crossrail line will open from 2019 or something providing an excellent new route.
For completeness, London City is the closest to the centre of the city, and popular with business travellers. It has direct "light" rail to both financial centres, taking about 20 minutes and with an excellent view. The three other London airports (LGW, STN, LTN) are relatively far away from the centre, LGW has the best rail connections, STN is in between, and LTN is a little annoying (there's a 5 minute free shuttle bus from the airport to the near-ish station).