I think the word 'puppets' goes too far. It assumes that our free will is taken away. However, the Moon does affect us. It does far more than just affect the tides. We are mostly made up of water ourselves. That is, our physical bodies range from around 90% water when we're first born, to around 70% as adults,which gradually tapers off percentage-wise as we grow very old.
So if there is a pull on the oceans, there is a lesser pull on us. But the effects of that pull...? There must be some but I'm not a really knowledgeable in that field. Certain any changes in our internal pressures, no matter how minor, would have a physical effect and, as our bodies affect our minds, so our thinking, maybe moods would be affected.
The word 'Lunatic,' a very old word, comes from the belief that certain cycles of the Moon affect people to the point where various sorts of mental instability get worse.
We need to remember that everything in the Universe affects us in some way, greater or lesser. The Sun is the obvious example: Day, night, hot cold, summer winter. But I'm pretty sure that there are radiations beyond our ken from the planets and, for that matter, the stars of the Milky Way. We've heard of gamma rays and cosmic rays. The fact that Astrology preceded Astronomy by thousands of years is no fluke of the intellect. People have been fascinated by the influences of our nearby neighbors in space from the Dawn of Civilization and possibly even before that.