This has a lot more to do with consumption than with production. Most of these islands are small, and it's unrealistic for them to have large populations or a heavy tourist industry without desalination plants. Energy needs to become cheap, and honestly, wind farms and solar can help with this.
I'm the article they point to a lot of factors as to why this is now becoming an issue: increase urbanization, increased industries, increased tourism.
They do say collecting rainwater is probably part of the solution, but if the other things keep on growing that won't always be the case.