Basalts are large rocks which can't be moved or destroyed. They occur frequently in later worlds of Adventure Mode or sometimes as part of a Set Piece in Sandbox Mode, and can come in the form of a pillar.
Basalt was added as a placeholder for Sinkholes. Upon the release of the Underground update, all Basalt in Sandbox Mode were to be replaced by Plugged Sinkholes.
Apart from decoration or forming an indestructible blockade (similar to the Tree Clump) it serves no other purpose.
Players can use the Imprisoned Pig Set Piece in which Basalt spawns to trap any kind of mob, even giants in Reign of Giants, and hold them prisoner there without fearing their escape. An easy way to trap them is craft a Telelocator Focus inside the "prison" and then use the Telelocator Staff on the desired mob. But using the staff near the trap is not advised, because the sanity drain would lower the obelisks, thus allowing the creature to escape.
A good use of this would be trapping a Varg and farm the Hounds it spawns periodically.