The blue stabilisers were a component in the TARDIS which stabilised it in flight and prevented it from rattling around.
The Eleventh Doctor believed the blue stabilisers had no purpose other than being blue. River Song used them to stop TARDIS shaking. The Doctor irritatedly called them "blue boring-ers," and said the journey was no fun with them activated. (TV: The Time of Angels)
At a point prior to the crash of the Byzantium from River's point of view, she wrote in her diary that she did not think the Doctor had ever seen the stabilisers. (GAME: The Eternity Clock)