So, for example, if a business in my neighborhood has a sign that says "No hippies, Jews, or N?????s", I should continue to shop there until they get convicted by a court?
It's probably worth asking the owner whether it's true, and if so, let them know that's why you don't want to shop there any more.
Just trusting an allegation alone and walking away silently seems both less fair and less effective.
Also, sometimes you find out more to the story. Maybe the owner is very old, and it happened a long time ago, and he's since changed his attitude dramatically. Maybe you forgive him or maybe not, but then you at least put it in perspective.
In any case, certainly don't repeat the unsubstantiated allegation to others.
What if the situation was, that someone told you they saw this sign one time? Would that be enough not to shop there?