It appears that when astronauts walked on the moon, they were very prone to losing their balance and falling over and finding it hard to get up.
So why is this? It turns out that being able to accurately judge which way is up is crucial for maintaining balance and via Kiki Sanford, I found this paper that examines the relative contribution of the various factors that lead us to judge whether we are upright.
Establishing an “up” direction is a multisensory process that integrates information about orientation obtained from visual cues, gravity and the internal representation of the body . Gravity typically contributes about 20% to the perceptual upright (PU: the direction in which polarized objects, including such things as writing, trees and people, are judged as being the correct way up) with the remainder coming from visual cues and the orientation of the body . Many studies have estimated the threshold for detecting linear acceleration . Estimates of this threshold vary considerably depending on the methods employed  but there is a general agreement that accelerations along the long axis of the body above about 0.15 m.s−2 (0.02 g) are reliably detectable.
To assess how much gravity is needed to establish an up direction, we had participants view a highly polarized visual scene while lying supine on a human centrifuge (Fig. 1a). We rotated the centrifuge at various speeds to create controlled, maintained linear accelerations along the long axis of the body (Fig. 1b). The visual scene they were viewing could be rotated about the naso-occipital axis, which had the effect of pulling the perceptual upright away from the body’s axis towards the direction indicated by the visual background. As artificial gravity is added along the body’s axis, there is a corresponding reduction in the relative influence of vision
We find that, even when the simulated gravitational force was above the acceleration threshold, it was only effective at influencing the perceptual upright above about 0.15 g: indeed, the gravitational force on the moon would only barely be able to provide adequate gravitational cues necessary for orientation.
Since the moon’s gravity is 0.16g, very close to the threshold of 0.15g, this means that astronauts are much more dependent on visual cues to maintain their upright posture. Given the barrenness of the scenery and the limited view allowed by their spacesuits, they could easily lose their sense of which way is up.
On the positive side, they would be able to stay upright far more easily on Mars which has gravity of 0.38g.