Starbase 51 was a Federation space station located near Trill and the Murasaki system.
In the year 2372, the USS Enterprise-E travelled to Starbase 51 for repairs to damage inflicted to the ship by a Changeling infiltrator. (TNG - Slings and Arrows eBook: The Oppressor's Wrong)
In 2376, eight Starfleet starships were operating out of the Starbase, including the USS T'Kumbra, the USS Sagittarius, the USS Polaris and the USS Appalachia. The latter was being refitted in August.
Admiral Leonard James Akaar contacted Starbase 51 in that month from Deep Space 9 to order the starships to engage and destroy the USS Gryphon. (DS9 - Mission Gamma novel: Lesser Evil)