A basement was the lowest level of a building where items could be stored for further use.
In 1930, Edith Keeler found James T. Kirk and Spock in the basement of the 21st Street Mission. Later, she commended them for doing such a good job cleaning it up. (TOS: "The City on the Edge of Forever")
Captain Jonathan Archer and the Akaali Riaan discovered a secret mining operations center including an antimatter reactor in the basement under Garos' curio shop in 2151. (ENT: "Civilization")
In 2375, Mila, Enabran Tain's housekeeper, hid Kira, Damar, and Garak in her basement on Cardassia Prime. They continued the tradition of cleaning the basement while there. (DS9: "The Dogs of War")