A cab from JFK or LGA is going to run you some $$$. A cab from JFK to Manhattan is $45 (flat rate), but that price doesn't include the tolls & tip, so it is more like $50+.
If you have time, your cheapest option would be to take public transportation. The bus from LGA is $2; once the bus gets into Manhattan you'd transfer to a subway to Columbia. The bus driver could tell you where to get off to make sure you'd get on the right subway. From JFK is would be a $7 trip - $5 for the AirTran, and $2 for the subway ride. You can actually take a single subway (the "A" train) from the AirTran stop near JFK all the way up to Columbia - it is a looong ride (1 hour+), but all you have to do is sit there. For $7, I would rather sit on the train than spend the money on a cab, if you have time.
Newark is an ok option too, but the NJ transit doesn't run as frequently as NYC public transportation, IMO. Plus it costs more to get to Newark and takes just as long.
Super Shuttle is always a good option too.
Good luck!