Stand the mattress up against an outside wall and vacuum out all the loose dust. When you are satisfied that all dust is removed, turn the mattress over and vacuum the other side.
Mix together a solution of 1 cup vinegar and 1 gallon of water. Scrub the mixture into both sides of the mattress with a clean wash cloth and allow it to dry.
Clean both sides of mattress with steam cleaner to make sure you kill off any small pests and draw any remaining dust or pollutants from the mattress. Allow it to dry thoroughly. You don't want mildew or mold within the mattress.