First of all - almost all of the adjustments to string height can be made with the truss rod. Turn clockwise to raise the fretboard (resulting in lower action), counter clockwise to lower it. The factory string height for our basses is about 3/32" or 2.4mm between the bottom of the string to the top of the fret on both the E and G strings at the 12th fret. The G string can be slightly lower due to the fact that it is the smallest string. Of course, tightening or loosening the trussrod affects the bow of the neck, which is how desired action is generally best achieved.
The factory specifications for the pickup height is 6/32" or 4.76mm from the top of the plastic pickup cover to the bottom of the G string. The angle of the pickup is set fairly level, so the larger strings are naturally a bit closer. If you haven't already adjusted the pickup or changed to a very heavy gauge string set, you shouldn't have to do any adjustments on the pickup.
On the 2-pickup basses the measurements are virtually the same on the bridge pickup and the neck pickup is further away, about 8/32" or 6.35mm from the bottom of the string to the top of the pickup cover.
On the new 3-pickup basses the measurements are (starting from the bridge pickup) 6/32", 7/32", and 8/32"; or 4.76mm, 5.56mm, and 6.35mm respectively.