What effect does the Moon have on the near Earth asteroid population? Does the big (0.012 Earth masses) Moon of Earth clear away NEAs, Earth orbit crossing asteroids, in a significant way? Venus and Mars don't have large moons, do they therefore have larger or smaller population of near asteroids than Earth would have if it were in the same orbit?

The outer planets have large moons, but also lots of captured asteroids, trojans and centaurs. Does a large moon even help gathering such objects, rather than ejecting them? Moonless Mercury and Venus seem to be pretty clean.
I think you answered your own question with your second paragraph. 

According to Newton's law of gravity 


the Gravitational force is directly proportional to the product of the masses and inversely proportional to the square of the distance between them.
So the biggest body in the neighborhood has the most pulling power provided it agrees with the equation.
As a consequence Jupiter and the outer planets should win out over Earth and keep them from hitting Earth or at least divert the collision course. Especially since the NEA come from their part of the cosmos.
I suppose the same logic would also work with the comets. 


Please see the following link for more:
http://earthsky.org/space/is-it-true-that-jupiter-protects-earth