Auto insurance is designed to follow the car, which means that the coverage you purchased for your vehicle will cover the car regardless of who’s driving it.
That said, in order to be fully covered by your policy (under what’s called the non-owned auto provision), your babysitter must have permission (verbal or written) to drive your insured vehicle.
Your babysitter must be an occasional driver only, not a regular driver or user of the vehicle. If they get into an accident while driving your car (and considering they had your permission to drive the car at the time of the accident), your auto insurance will be considered the primary coverage in any accident-related claim that's made. Your babysitter's car insurance coverage (if s/he's covered under his or her own policy, or that of a parent) will be considered a secondary coverage, and will only activate if damage claims exceed the limits of your own policy.