I just don't get this decision at all!
Forget for a minute the domestic concerns regarding US bases, or the geopolitical concerns of yet another potential blockade on Chinese shipping lanes, and look at this from an environmental perspective:
Jeju as a whole is now a registered UNESCO Natural World Heritage Site. That alone should be a veto from the UN should it not? Is there something I'm missing here?! The place the proposed base is to be built at is on the southern coastline which is basically a giant stretch of columnar basalt akin to the Giant's Causeway. They want to build on that. Oh, and the site is literally in the middle of the main tourist resort area on the island (no, seriously, look it up, dead centre between Seogwipo and that premium resort complex, look for all the waterfall icons on your tourist's map and scan around a bit).
From a pure economic standpoint even, I honestly don't know how they can justify crippling the island's hospitality industry for a single base.