Cheapest is by public transportation which includes subway/ferry OR subway/PATH train. Here you go: by ferry-take Airtrain from your terminal to Howard Beach Station. Take A subway to Chambers St. Walk west on Chambers towards World Financial Center/Battery Park City (located on the west side of the WTC site). The NY Waterway ferry terminal is at Vesey St. (south of Chambers), take the ferry there to Port Imperial/Weehawken. Ferry only runs AM from 6am to 9am and PM from 3:30pm to 8pm, trip takes 15 minutes. Airtrain/subway is $7.25 and ferry is $12. OR take Airtrain to Jamaica Station and take E subway to 34th St./Penn Station. From there take PATH train to Exchange Pl. and local transportation to Weehawken. PATH fare is $1.75.
Cheapest is by public