If stone is kept inside this is generally enough to protect it as it is hard wearing. Softer or more porous stone, particularly alabaster and limestone, can be damaged by moisture however so museums have to take extra care with these to make sure the environment is dry and watch for flaking or white salt growth.

This is a picture of marble bust being carefully moved under the instruction of specialist conservators at The Bowes Museum.
Stone should be kept clean of dust but using the wrong cleaning products can be very damaging as strong acids or alkalis can leave permanent marks and weaken the stone.