Depends on the capabilities of the familiar. Something that can trigger a wand of Unseen Servant can deal with the work part of babysitting an infant, and an intelligent animal can amuse the baby for hours. Once the kid goes mobile, however, you need somebody who can actually ride herd and deal with medical booboos; this is not most familiars, but may be some. If the kid is the offspring of the familiar's arcane caster, they'll have an emotional bond of their own based on the familiar's constant presence, and the empathic bond of familiar with parent will be better than any emergency number; so a familiar would make a good choice to watch the baby even when it's not the primary sitter.