Firstly, a bright moon can allow people to do things at night that might be difficult or impossible without the extra light.
Secondly, the changing phases of the Moon impressed many early cultures, affecting the calendar and religious practices.
In the last, the tides themselves affect access to tidal pools and the sea, so especially for coastal dwellers, tides can directly affect how they live.
All those sort of things being able to do things because of the light of the Moon, the tides it causes, and religious and other influences on life are about it. It does not directly affect on people that we know of. As people are animals, and many animals’ biology has used changes in the light provided by the Moon to regulate their biochemistry, and a a result, some things, such as the timing of ovulation in women, are similarly linked to the phases of the Moon. But that is a matter of evolutionary development related to the past and has nothing to do with any actual force by the Moon, so if the Moon ceased to exist, the only effects on humans would be alterations in first and last case.