I think the word you want is “credibility.”
There is the old saying about how if one man says you are a donkey, ignore him, but if two say you are a donkey, get yourself a saddle. Herman Cain backed out at 2 allegations, which were, interestingly enough, never pressed once he did. But this is a lot, and where there is smoke there is fire. Where the fire IS, may still be unclear.