a big sponge and bucket of water mixed w/stone safe cleaner.

or you could do an old fashioned type of mop.

I'd prob do the sponge first to get all the dust, and then finish it off w/the mop

How old is the floor? Assuming it's concrete dust that you have, I'd look into an acrylic concrete sealer.

Yes, it needs to be sealed.

House is probably 60+ years - will try sponge/mop washing first. Thanks!

Mopping will only last until the floor dries out and then the dust will come again so sealer is the answer I assum this in an unfinished basement so what is the reason for going into it often.

Yes, it is unfinished but I walk through it from my garage, have laundry and a work out area. I do have rugs down for walkway but still have to walk on concrete in garage area.

Park your shoes at the door and wear slippers in the house.