Originally Posted by PilotFrog

JFK-EWR
Air Train to Jamaica Station
LIRR Jamaica st - Penn Station
NJ Transit Train Penn Station - Newark Airport (This is the toughest leg of the journey, hard to time the trains that stop at Newark Airport)
Air Train to Terminal B
Says it takes 1:56, but that is assuming you hit the trains right, and that is best case.

You can also do
Air Train to Howard beach
A train to Canal St
Path to Newark (city)
NJT train to Newark Airport ( again this is the tricky part )
Air Train to terminal

JFK-LGA
Not sure, but by public trans you can
Air train to Jamaica
E train to Sutphin
7 train to Archer
Q48 bus to terminal

There is also the Q runner, but not sure how that works.