What kind of stone are we talking about? I have slate and travertine in my house and neither gets dusty? I am curious about what is going on here? When you say dusty are you literally just talking about dust in the air settling on the floor, just like it would on your piano, or TV, or entertainment center? Or is the stone dusting, like concrete can do?
I use a kenmore bagged canister vacuum with the hard surface floor attachment for vacuuming my stone, tile, and hardwood. I damp mop my floors a couple times a week at least (this is very quick, I can do the kitchen, halls, entry, etc in 5-10 minutes) and I hand scrub the kitchen and baths maybe twice a year if they need it.
Lowes and HD both carry good sealers. Most likely a penetrating sealer is what you want but it would be good to know what kind of stone this is. The more expensive it is the better it is.