The best way of cleaning a gravestone safely without damaging the gravestone is to use some dawn dish soap or other dish soap in a bucket of water.
Dip a rag or sponge in the soapy water and gently scrub the gravestone to remove an dirt or grime from the gravestone.
I've always used dawn dish soap and water to clean gravestones and never has it harmed the gravestone.
Just do not use any harsh chemicals or things such as vinegar since the vinegar is acidic and can damage the gravestone.
You can also use some Purple Power degreaser on the gravestone as it's non harmful to the gravestone and is an safe cleaner.
The purple power degreaser works very well on a variety of surfaces as well and is a wonderful product.