Take one of the various public transit options to Manhattan (most likely either train to Penn Station or bus to Port Authority or Bryant Park), then take the subway to Brooklyn. Bedford Ave is very long, so the choice of subways will depend on where on Bedford Ave and what day of the week you're going (the L, J or M for Williamsburg; A for Bed-Stuy; 2, B or Q for most other parts of Bedford Av).

Unfortunately there are no public transit options direct to Brooklyn from EWR.