Cheapest by far is by subway/Metro North rail. Take AirTrain ($5) from terminal to Jamaica Station, transfer to E subway ($2.25) to Lexington Ave/53rd St, transfer to downtown #6 subway for one stop to 42nd St/Grand Central, at Grand Central take the Metro North-Harlem Line train to White Plains ($7.75 off-peak, $10.50 peak) and then take the #12 Bee Line local bus from the Transit Center to Lincoln Ave/SUNY Purchase.
Cheapest by far is by