I've always believed that the phases of the moon do indeed impact many things; earthquakes included.
I believe the phase of the moon affects the Nenana Ice Classic. My pick below, if I were to buy a ticket.
People believe a lot of things.
The overwhelming weight of carefuly statistical studies finds no moon phase correlation whatsoever, particularly with large quakes.
In certain circumstances (volcanoes, undersea spreading ridge quakes, etc.) there is a potentially statistically significant correlation with diurnal (daily) tides. The changes in stress from those are far greater than the changes in the sizes of tides around full and new moons (and the idea that there's something special about full moons as opposed to new moons and earthquakes is flat out silly.)
Basically, when you have a quake near a full moon, people start babbling about it being near a full moon. For any of the many large quakes nowhere near a full moon, the same people are silent.