I'm sure Jesus would have enough brains to know the difference of spending time preaching with harlots and having them bunk with him and travel for long hours with 12 other men. Its like letting your son go over to a girls house and then not letting him sleep over. Why? Because even though you trust both it's completely unethical socially. It's not totally about reputation, its about not giving other people the fuel to make a fire.
but we are not talking about what Jesus knows you were referring to what people would think of Him so you are saying that having a whore wash your feet would not give people fuel to make a fire . only if He had let a woman travel with them, such as mary and martha and any other of the sixty people that may have been Women that were disciples but were never giving a leadership role in the Church after He left,