Luna certainly affects life on Terra. Everything from the tides to wolf hunting behavior to human menstrual cycles are influenced by its orbital period, primarily through gravity and reflected light. It's not hard to think of good reasons why. Considering how light affects human actions, it makes sense that the greater nighttime illumination offered by a full or gibbous moon could change how people act.
I have my own hypothesis for how full moons and strange behavior became associated. In pre-industrial societies, Luna was the only major source of light after sunset (and candles really didn't do much for interior lighting, either). When we think about how much time agriculture and other forms of manual labor take up, those people would be thankful for the extra hours afforded by a full moon.
The thing is, staying up later makes people tired, and tired people do foolish things. Ergo, people are crazy around a full moon--both before and after.
In our modern age, however, we don't spend much time looking at the moon. In fact, most of the light that keeps us awake comes from television, phone, and computer screens. Perhaps the decrease in that superstition can be traced to the advent of television, or all the way back to widely-available lightbulbs.
Either way, I'm turning the laptop off and going to bed.