Hi. My husband and I went to Grand Cayman on the voyager a little over a year ago for our Honeymoon. We did not have an excursion arranged, but decided to walk to locate 7 mile beach on our own. Getting off the tender we went left, and walked a few blocks or so. We then saw an opening so we crossed the street and combed the beach. Their were a few places you could do this. We walked along the beach for a couple hours stopping and playing in the water some. There was one spot that we took our shoes off and walked though the water to keep walking along the beach. You will see a path were many others have done the same, we always felt safe along this beach/area.
On the way back we walked on the sidewalks on the street side, and did a little shopping. Later on we had chosen to walk the beach line again. Along the street side there were convenient stores and fast food restaurants to stop and get drinks if needed. Taxis drivers were real aggressive, slowing down encouraging you to have them take you to beach. Even when we were practically there already. I would do this island by your self, it was perfect for that.