In an emergency near a remote island is it better to land resulting in the airport closing or ditch in the water?
In general, it's a bad decision to ditch if there's an alternative, even a gear-up landing (Sully ditched in the Hudson because the alternatives were much worse: virtually certain total loss on board, plus significant ground casualties due to dense population).  The safety of the crew, passengers (if present), and people on the ground is the first concern of a captain or PIC.  Ditching seldom ends well, and often results in 100% fatalities, because even with a perfect ditch, rescuers often can't reach the aircraft quickly.

For a sufficiently remote location that incoming flights may not be able to return or divert (like the example, Easter Island), there may be an option to land alongside the runway (potentially a better choice for gear-up anyway, in a location without a large fire crew, since grass is less likely to start a fire than tarmac), both for safety and to keep the runway open.