From pictures taken on the moon, it appears that there are no stars visible in the sky, but I do not know if this is an effect due to cameras.
Quote from Buzz Aldrin:
“Obviously, when we touched down on the moon, we were very relieved. The immediate surface was very powdery, as best we could see looking down from 15 feet. Off in the distance was a very clear horizon, maybe with a boulder. And, of course, the brightness of the sunlit surface was almost like looking out at sunlit snow. Your pupils close down, just as in orbit when the sun is on the spacecraft. The sky is black as can be, but there’s no way you can see stars. They’re there, of course, but you can’t make them out because they’re too faint with all the ambient light in your eyes. I was impressed with the total “magnificent” desolation I saw in front of me: black sky, no stars and the horizon clearly curving away.”
That's because those photos are showing the black backdrop of the studio where they were taken...kidding.
The stars are not visible on the Moon, at least in photos, because the lunar surface in sunlight was so bright that the stars were washed out. No camera (nor the human eye, usually) has the dynamic range to process something very bright next to something less bright. If the aperture were open wide enough and the exposure long enough to collect enough light to image the dim source, it would collect so much light from the bright source in that time that this latter light would overpower everything else, and you would not get a comprehensible image. Try to take a picture of a candle flame with the mid-day Sun in the background.
As for human eyes, astronaut testimony may confirm stars were visible on the Moon when looking away from the surface. I do not have that information. I tend to think if a camera were pointing straight at the sky with "blinders" on to block surface light, you would see the stars just as well as in empty space.
The thing to remember on the Moon is, the lack of atmosphere means there is no refraction or scattering of light. The only light you see in any direction is the light that came from that direction. If the Sun is in your field of view, you see it. If you turn your back to it, the sky is as dark as night. Anywhere around the Sun is also as dark as night (though you may not see stars, as mentioned before). When the Sun sets, its light immediately disappears, though you would still have light from the Earth, so it may not be pitch black, but would be much darker.
This was a deliberate choice by the mission planners to avoid the obvious problems of landing at night i.e. it’s dark at night time.
The reason you can’t see stars on the Moon during the daytime is the same reason you can’t see them on Earth, your eye is adapted to the average luminance in your field of view.
The day lit Moon’s surface has a more or less constant luminance of about 2.5 kcd/m2 the Earth varies depending on cloud conditions between about 0.7 and 5 kcd/m2. A moonlight night on Earth is about 1.4 mcd/m2 - note the k for kilo and the m for mili; we’re talking 6 orders of magnitude here.
The human eye is good, it can adapt to cover about 10 orders of magnitude: just not all at once. Once adapted, it can cover about 3 orders of magnitude. That means, for an eye adapted to look at the Moon or the Earth, the stars are just too dim.
On looking from the moon the sky appears black as there is no atmosphere to do Rayleigh scattering. Rayleigh scattering is the scattering of light by particles in a medium, without change in wavelength.
