I'm surprised the idea of a familiar baby sitting is so odd. I mean, who looks after the wizard normally?
My alienist wizard, Lethey, has a wisdom of 6 and an charisma of 8. Without prompting from his telepathic and 'tongues' enabled pseudo-dragon, even the most desperate patron would hesitate to turn to him for help, and he'd get ripped off by every shopkeeper and tavern-owner on the route.
Prestidigitation is by far the most useful hygeine spell in the game. Clean diapers without having to take them off. With the spell-linked familiar feat, he can cast it once a day for an hour, he has as many ranks in heal as the wizard does when it comes to recognising symptoms, he's just as strong as the wizard is when it comes to picking the little guy up. True, he has a problem with doorknobs, and he tends to fumble the bottle.
He's not really the ideal babysitter, but compared to your average professional adventurer, even active neglect starts looking pretty good.