The Second Doctor when offered a "bacterial check" indignantly argues, "I'll let you know that our TARDIS is as sterile.... " The Eleventh Doctor, on the other hand, when the Silurians attempt to decontaminate him, cries out, "Remove all human germs, you remove half the things keeping me alive." So possibly the Doctor doesn't need to wash his hands after a long day spent traveling through time and space, because the TARDIS keeps them sterile for him or because washing them would remove symbiotic bacteria or because GALLIFREYAN PHYSIOLOGY and we should just let this line of inquiry go. Right. Something we do know for sure? His human companions still need to wash their hands. Convincing adults that hand washing is necessary is usually pretty easy: the CDC informs us that hand washing can reduce the risk of respiratory infections by 16%. There you go. Science. It can be hard to convince children, though. Let The Doctor help. No, not your pediatrician. Lollipops won't fix this situation. The Good Doctor. One look at this TARDIS, and they'll be ready to wash whenever you say so. But make them try to recreate the VWORP noise when they pump it. That's fun for everybody.