Byzantium still doesn't care about the 2nd crusade, the 3rd may see Barbarossa press ahead to crossing Anatolia in late autumn (instead of events leading to him winding up waiting until spring), the 4th probably sees a lot less interest in going to Constantinople - and the rest didn't make much difference to the Byzantines.
Assuming no butterflies on any of those crusades.
As for relations with the other powers in its neighborhood, it might change things involving Roger of Sicily (around the time of the 2nd Crusade), but it might not.
I don't want to say its completely irrelevant, but I think the main western powers which disliked Byzantium because of crusader related stuff aren't going to really warm up to it anyway, even if this goes better.