It's sole current use is to be made into Robust glass. Like the stone itself, the glass is also untradeable. The special glass likewise has only one use currently; to be made into Potion flasks, a tradeable item which can be blown from the glass at 89 Crafting.
Each player can mine 50 red sandstone per day. Jagex intends that the limited supply and high demand for these flasks mean those who have the Crafting and Mining levels to make the new potion flasks will make a healthy profit (if you choose to sell them rather than keeping them for yourself, that is).
The normal amount of 50 red sandstone can be increased in 2 ways:
Like some other once-daily content, the red sandstone "day" resets at midnight, British Standard Time, so one can essentially mine 100 pieces every second day by starting just before reset time. Exiting to the lobby and reentering the game may be necessary to reset the player's instance of the red sandstone mine after the actual server reset time.
Interestingly, unlike most items that cannot be traded, Red sandstone can be placed into a Beast of Burden, which means it is possible to mine all 50 stones with a Pack yak familiar without banking.