Most of the LOBs trips that we have done, including the Red Sea Aggressor, stayed at sea the entire time without opportunities to visit sites on land, but there have been a few exceptions on some Caribbean and Bahamian LOBs.
The Caribbean Explorer departed from St. Maarten and we dived around St. Kitts and Saba. On 2 afternoons we were given the option to either keep diving or to take a panga to shore and explore those 2 islands. We choose to explore the islands and we greatly enjoyed the chance to see a bit of them.
When we were on Blackbeards long ago, it cruised from Miami to the Bahamas and one afternoon we docked in Bimini and had the opportunity to enjoy the amenities of that island - it was very quiet and undeveloped back then but we had a lot of fun. One night the Blackbeards boats docked off of a small island occupied by "sand miners" working for a glass factory. We had a bonfire and barbecue party on the beach with the crews and passengers from the other boat and with the miners.
We also spent some time in Bimini and Freeport when we were on another Bahamian LOB, perhaps it was Nekton? I know that the goal of a LOB is to "eat, sleep, dive" but we've also appreciated opportunities to go onshore and look around a bit.