Leonard Bernstein was a social activist in that he embraced many liberal causes and, once he was famous, tried to use his fame to promote those causes.
Bernstein is most famous, or perhaps infamous, for hosting a fundraiser at his home that was seen as a benefit for the very controversial Black Panther Party. Bernstein and his wife felt that the point of the fundraiser was to help promote civil liberties rather than to promote the particular point of view of the Black Panthers. However, the general public perceived it more as an expression of support for the Panthers' violent and nationalistic viewpoint.
Bernstein was also involved in other liberal causes. Even before he was famous he was interested in pushing for workers' rights. His senior thesis at Harvard explored the importance of black music in American music in general. He would continue to push to advance blacks and women in the music business. His best-known popular work West Side Story, can be seen as a political and social commentary.
Bernstein, then, was an all-around liberal activist who felt that he could and should use his fame and his music to promote social change.