Examine the stone for signs of deterioration such as chips and crumbling. You should not clean a stone that is already deteriorating since cleaning may hasten the process.
Determine the type of stone. Some stones are softer than others and therefore more prone to damage. Limestone and sandstone are particularly susceptible to harm, while granite is much tougher, so choose scrub brush bristle stiffness accordingly.
Wet the surface of the stone and scrub using plain water with the softest bristle scrub brush that will do the job. (Never use wire brushes!)
Rinse the stone often with clear water as you scrub. According to Jonathan Appell at Gravestone Preservation, a pump sprayer is better for this purpose than dipping the brush in a bucket of water because it reduces contamination from dirty water. He also suggests starting at the bottom of the stone to prevent streaking and staining; and using a non-ionic (neutral pH) detergent and biocide to remove deep-rooted biological growth that may contribute to a stone’s demise.
Allow the stone to dry; then treat with a consolidating material (NOT a sealer, as that will not allow the stone to breathe and will exacerbate deterioration). Jonathan Appell recommends a product called “Conservair”—particularly for sandstone. Follow the manufacturer’s instructions for application.