Hilmar already posted a good answer above, but I will suggest a modification that doesn't fit in a comment.
[Arriving early is] clearly a sign of a responsible and reliable babysitter. These
are in my experience very valuable and hard to find. While you are
legally not required to pay her for the extra time, it would be
stupid, not to. The extra half hour is probably just a small fraction
of your overall expense for the night.
Rather than "paying for the extra time", I would be inclined to frame this as an increase to the hourly rate: "We notice that you take trouble to make sure you're always here on time, even if that means leaving early, and as a sign of appreciation for your reliability we'd like to bump up your hourly rate."
If you like, you can make sure the raise works out equivalent to paying her for the extra time. For example, if she usually works for four hours, then instead of adding half an hour to the calculation you can add 12.5% to the rate. It's the same money either way, but I'd argue that it's better to do it as a change in rate.
Her showing up early isn't useful to you in itself; it's just a means to an end. The valuable part is knowing that she will be there by the time you need her. If you pay her for arriving early, you create an incentive to arrive even earlier, which again isn't helpful. But if you position it as a reward for a responsible, dependable attitude, you encourage responsibility and dependability, which are always desirable.
It's also better preparation for future work. Although there are some exceptions, employers don't usually pay for travel time.