I did an internship at NASA where I worked on the lunar dust problem. A big part of the problem is that the dust gets charged by solar radiation since the moon has no atmosphere. During my internship they had me read reports from each of the moon missions discussing how the astronauts dealt with the dust. What they basically ended up on is opening the door on their way back to earth so that the vacuum sucks the dust out.
It claims the dust is likely charged by solar radiation, but doesn't say the polarity. The paper mostly talks about using "dielectrophoretic forces"–forces experienced by any dielectric (polarizable) particle in a non-uniform electric field regardless of its net charge.
It is likely that most particles become charged positively, after the incident radiation expels electrons from them.
For negative charges, electrons must be captured. There is a minor component of the cosmic radiation that consists of electrons and it is also possible to capture photoelectrons emitted by nearby particles, but in most cases there should be much less opportunities to capture electrons than to emit them.
Even an incoming electron from the cosmic radiation would first expel a great number of electrons from dust particles before slowing enough to be captured by some particle.
EDIT:
As explained in the document linked by another poster, what is written above is correct only for the day side of the Moon.
On the night side, electrons are no longer emitted, but only captured, so the sign of the charge is reversed.
I could not help but imagine with a laughter that finally technology will keep houses clean with seamless effort and low cost. Note: my high-end cleaning robot is dysfunctional.
I was wondering too, if maybe under the right conditions, humidity sensors could trigger polarization of the glass surface followed by opposite polarization could potentially rid the surface of at least some of the dust. Some ionizing devices even produce a 'wind' of sorts.
