Since the moon is tidally locked and has the same face showing to the earth, I would expect that the earth's shadow represents an area on the moon that would be spared some meteoroid impacts. Does the side of the moon opposite the earth have more craters than the side facing us?
To directly answer your question: Yes, the far side of the moon (common term for the side not facing Earth) does have significantly more craters, so long as your minimum crater diameter you're counting is several tens of kilometers or miles, or smaller (approximate enough that metric/imperial doesn't matter much here).
However: That answer doesn't really get to answering the bulk of your question. Crater spatial density (number of craters per unit area) is a proxy for surface age, where the more craters you have, the older the surface is. Given the many, many more craters on the lunar far side, the reason is that it is older. The near side has been resurfaced much more recently than the far side by the large craters that were later filled in by lavas. Those are what you see now as the lunar maria, or dark "seas." Now, with that said, one of the fundamental assumptions of using impact craters to date surfaces is that the craters form as part of a roughly uniform distribution over long time periods and they form with a uniform spatial distribution. So, we can't actually use the craters yet to say that there are more on the far side because it's older.
Modeling: Several different researchers have modeled all the effects of the Moon's orbit around Earth, the Earth-Moon system's orbit around the Sun, and the trajectories of objects that would hit the Moon. I think there's a more recent one, but the most recent off the top of my head is Le Feuvre & Wieczorek (2011). What they found is not that there should be an enhancement on the far side, but that there should be a marked enhancement on the leading side, which is the western limb (it's the side on the left as seen from Earth).
Why: Due to how the velocities all work out, the Moon is always moving forward in its path around Earth / the Sun. It never actually back-tracks during half of its orbit against our orbit around the Sun because the orbit around the Sun is faster than its orbit around Earth. Because of this preferential motion, it is plowing through solar system debris, and so that leading edge plowing through the solar system will accumulate more, larger craters (larger because the impact velocities will be a bit higher).
But: Many different researchers have looked for this effect in the cratering record and have not found any clear signal of it (or, for every study that finds a signal, there's another that doesn't, or, the results are in the noise). So, while modeling is all well and good, if there are not observations to support it, there's always the question of whether the modeling is correct.
Summing Up: Based on models that factor in a lot of the different mechanics going on, they all pretty much say that the leading edge (around 270°E / 90°W) of the Moon should receive a higher impact rate than either the near or far side. However, actual evidence of this effect is hard to come by, so while there's no good reason to say the models are wrong, they cannot yet be confirmed. If the models are correct, the effect is small enough that the difference between expected craters on the near vs far side is too small to be visible in the crater counts, so the far side does have more craters because it is an older surface.
Do Lunar Maria count as craters or not?
They are low lying areas which were flooded by dark colored lava which dried, covering up previous craters. And some of the low lying areas flooded by lave are large impact basis produced by the impacts of large asteroids.
Thus some people might count the large impact basins as giant crater, and might consider that the lava flows have covered uplarge lu numbers of craters. And the maria are mostly on the near side of the Moon.
Since the Earth has 81 times the mass of the Moon, Earth's gravity tends to attract asteroids toward it. If asteroids fall straight toward Earth, the far side of the Moon would intercept a small percentabe of the asteroids falling toward Earth and would have craters, and the near side shouldn't have craters.
Asteroids which pass close to Earth would have their orbits changed and then move off in varius directions.
Mentions that some asteroids passing close to the Earth might be diverted into paths leading to the Moon.