The results of some experiments done with mice on the ISS have been published recently and they cast doubt about the possibility of living for a long time on the Moon.

For mice, at least a third of the Earth gravity was required to prevent serious health problems and at least two thirds to avoid any health problems.

So it seems that the gravity of Mars is close to the minimum compatible with long-term staying, while that of the Moon is insufficient. Therefore people would have to spend only a limited time on a Moon base, much like on the ISS.