Seven Mile Beach is easy to get to because, well, it's seven miles long. We spent a week in Cayman a few years ago and enjoyed every morning on that beautiful beach.
You can either take a beach barbecue excursion from your ship, or simply take some cash with you, grab a taxi and have them drive you to a good beach access point. They'll know the best one. It's not a long drive at all.
You can also get to the beach through one of the resorts, like the Marriott, the Hyatt or the Westin. They'll have restaurants and bars nearby where you can drink and dine.