Note: The question and answer below deal with lunar "earthquakes" caused by the tidal stretching of the Moon by the Earth, and the nonexistence of similar earthquakes caused by the tidal stretching of the Earth by the Moon. However, there are at least four known causes of moonquakes, and not all are as minor as those discussed below.
The seismometers left on the Moon by the Apollo astronauts recorded nearly 30 moonquakes during the five years they were in operation. Some of those were relatively weak, but some were surprisingly strong -- in fact in some ways far stronger than similar quakes on the Earth. One type of quake believed to originate at depths of up to 700 miles, involved fairly weak tremors due to the tidal forces discussed below. Another, occurring near the surface, was probably due to meteoric impacts. And a third, even weaker but still detectable, was probably caused by thermal shock due to the sudden heating of the surface at sunrise.
The biggest quakes were shallow quakes of unknown origin (perhaps due to landslides), which had intensities estimated at up to 5.5 on the Richter scale. That would be strong enough to be of concern to future lunar colonists even if lunar quakes behaved the same as those on the Earth. But they don't. Unlike earthquakes whose vibrations usually die out in a few tens of seconds, lunar quakes can go on for as much as ten minutes. The reason for this is that on the Earth are weathered by wind and water and altered by geological forces. This makes the rocks somewhat "soft" and causes them to absorb the vibrations passing through them. On the Moon the lack of weathering and geological activity makes the rocks very hard and stiff. As a result they "ring" like a bell, and quake vibrations can bounce back and forth inside the Moon for a very long time before finally dying out. This means that future lunar bases may have to be able to withstand strong vibrations for very long periods of time.
Query: What is the effect of the Moon on earthquakes?
Although there are a number of popular myths connecting earthquakes and the Moon, careful (statistically valid) studies show that there is no correlation between earthquakes and the time of day, the position of the Moon in the sky, the phase of the Moon, or the distance of the Moon from the Earth. On the Moon there is a relationship between the distance of the Moon from the Earth and "moonquakes". Most lunar quakes originating in the deep interior of the Moon are caused by tidal stretching of the Moon, which averages about 30 feet (about 20 times greater than the average Earthly tide of a foot and a half), and can increase or decrease by about 20% of that 30 feet (or about 6 feet) due to changes in the lunar distance.
The math works out as follows:
Earthly tides average about 1 1/2 feet (based partly on the theory of tides and partly on observation)
The Earth and Moon exert the same overall gravitational force on each other (Newton's 3rd Law of Motion, the Law of Action and Reaction)
The Earth is about 80 times more massive and four times larger diameter than the Moon (observation)
Reasoning and Results:
Since the Earth exerts the same force on a body 80 times less massive, it would produce 80 times larger tides on the Moon than the Moon does on the Earth, all other things being equal. But tides are caused by tidal forces, which are the difference between the gravitational force at a given place and the force at the center of the body (which equals the average force on any part of the body), and since the Moon is four times smaller than the Earth, this cuts down the effect of our larger force per unit of mass by a factor of four. Hence the final result that lunar tides average 20 times larger than ours, or about 30 feet.
Unlike tides on the Earth, which move eastward on the Earth as the Moon moves westward in the sky (thanks to the rotation of the Earth), lunar tides do not move around the Moon, because the Earth doesn't "move" around the Moon. Since the Moon always keeps one face to the Earth, the Moon experiences "frozen" tides in which the tidal stretching (which always points toward and away from the Earth) is greatest at the centers of the near and far sides of the Moon.
If the Moon kept a constant distance from the Earth the tidal stretching would be fixed, as well as "frozen"; but it gets up to 6% closer to or further from us than on the average as a result of its orbital eccentricity. This increases or decreases the overall force by 12% (since gravity is an inverse square law), and the tidal force which stretches the Moon by 18% (because the Moon looks bigger when its gravity is stronger due to being closer). Hence the approximately 20% change (orabout 6 feet) in the size of the lunar tides over the course of a lunar month.
As a result of the 6 feet / 1000 mile stretching, the Moon suffers weak earthquakes at about half the depth of the lunar radius, which are strongly concentrated around the time of its closest approach to the Earth (perigee), every 27.3 days.
Now one could argue that if the Earth can do this to the Moon, the Moon ought to be able to do it to the Earth as well, but as it turns out it cannot, for two reasons:
(1) As stated, the force on each body is the same, but the tidal force/stretching on the Moon is 20 times larger than that on the Earth because of the difference in their size and mass. If we were to ask what kind of seismic activity we see on the Moon when the stretching is only 1/20th of the maximum and is equal to the typical tidal stretching of the Earth, the answer would be: none.
(2) Also, tidal stretching does not occur at the surface of a body, but throughout its interior. In that part of the Moon which experiences "moonquakes", the overall depth is more than half of its radius of 1000 miles. Hence, over 50% of the tidal stretching occurs there, or about 3 to 4 feet of stretch in 500 to 700 miles. That is what is causing the moonquakes. In the case of the Earth the overall tidal stretching is 20 times smaller, but the area where earthquakes normally occur -- the lithosphere -- is only about 1% of the Earth's radius (about 40 to 50 miles, out of 4000 miles), so the stretching in that region is only 1% of 1 1/2 feet, or 1/2% of the lunar stretching over a distance over ten times smaller. Correcting for the smaller region the average stretch within the portion of the Earth subject to normal faulting is only about 2% of the stretching that occurs on the Moon when moonquakes occur.
In other words, although the powerful tidal forces exerted on the Moon by the Earth can easily cause weak quakes in the lunar interior, the much smaller stretching of the Earth by the Moon in its crust/lithosphere is far smaller, and cannot be expected to exert any influence on Earth quakes.
That doesn't mean, of course, that it isn't possible for some subtle mechanism to amplify the lunar tidal effect, and cause earthquakes to be somewhat correlated with some aspect of the lunar position(s). So many studies have been done looking for such correlations. And as is true of any large group of studies, some of them should yield apparently meaningful results, just by chance (e.g., if there is a 1 in 25 chance that an apparently significant result would result by chance, then 1 in 25 studies will have an apparently significant result even if there isn't any real connection). Needless to say it is the "chance" studies that are trumpeted in the press, hence the general misconception that the Moon has some effect on earthquakes. But if such an effect were at all real, geologists interested in predicting earthquakes would gladly use it. So there have been lots of studies, all of which (save for the occasional random chance "false positive") show no statistical correlation between (1) the position of the Moon relative to perigee (there should be more quakes when it's closer, but there aren't), (2) the phase of the Moon (the solar and lunar tides add up at full and new moon, so there should be more quakes then, but there aren't), (3) the position of the Moon in the sky (rising or setting, tidal forces might compress the crust, while overhead or underfoot, they might stretch it; the effects are unknown, but if significant should yield non-random results; and they don't), (4) and so on and so forth.
In other words, though both theory and observation demonstrate that deep lunar earthquakes can be caused by the effect of the Earth on the Moon (though only near perigee), they also show that there is no observable or expected effect of the Moon on Earthquakes.