A quality pair of hiking boots is recommended. The hike to the grottos is relatively short and easy, and worthy of several hours of your time. 
What brand hiking boots work for a woman's narrow foot?
I would wander into any REI, the footware people tend to be extremely knowledgeable and able to fit people feet into anything. Give yourself at least an hour to two to try on boots and make sure to bring the socks you'll be using on your trip.
I have found that REI brand and Merrell boots tend to run a little narrow, and tend to fit a good amount of people. If your foot is slipping foward, that just means that the boot isn't laced tightly enough, since that happens when the top of your foot can slide around. Try cinching down whatever boots you have a little more, possibly even a thicker sock. Another option would be trying after market insoles, like Superfeet. These tend to have much better arch support, and will boost your foot up inside the boot a decent amount, making it a snugger fit.