I think he was cowardly to name names and as such a huge talent that he was, would his career had died if he had refused? Of course not. I think even in the 40's and 50's, no one in the theater or ballet cared if you were gay. Of course that is 2nd guessing at this point but he did ruin some lives by naming names, as others had done. (I remember my parents not allowing me to watch The Ed Sullivan Show because of his political views, although they relented when The Beatles appeared).
I'm 52, so my memories of the HUAC hearings and their context are second-hand, but I've always felt that the people called in front of that committee were hung in the middle -- damned if they named people who weren't already "out" and damned if they didn't. While it's true that Robbins' immediate colleagues in the theater wouldn't really care if he were gay, straight, bi or otherwise, the next layer out in the community, which would include funders, might care, or care if the general population thought less of them for being associated with someone who wasn't "normal." Remember, it wasn't until 1973 that the American Psychiatric Association removed homosexuality from their list of "mental disorders." If I'm seeing the color-coding on the map correctly on the Wikipedia page on sodomy laws, the only state that had repealed them before 1970 was Illinois. Sadly, Robbins lived with a real threat at that time.
There's all kinds of speculation about what might have happened if someone who was truly popular, as Robbins was, stood up against the committee, and I'm sad that it will only ever be speculation, but I do feel truly sorry for him, and so very grateful that I've not been in a situation that dire.