Not to say they shouldn’t be taking measures to prevent spread (obviously incredibly important on an island), but does the airport have the authority to do that? Seems like something the mayors or governor would have to do?
Correct, the feds control the airport.
The intent of the home/hotel room quarantine LG green is proposing is to discourage anyone from traveling to Hawaii for fun and also to discourage Hawaii residents from leaving and potentially returning with the virus. The problem is that there is no practical way to enforce the quarantine. They can't station a national guard member outside every hotel door, apartment building, or house. Thousands of people are still arriving every single day.
What the governor and mayor CAN do, however, is to order everyone to shelter-in-place. Every visitor would need to stay in their hotel room, only take-out food would be available, no activities or shops would be open, beaches and outdoor activities would be closed . . . who would want to come? Even if people did come, they'd have limited contact with the local population, further reducing (but not eliminating) the possibility of spreading the virus.
I think we're on the brink of a shelter-in-place situation here. Maui is "highly encouraging" people shelter-in-place, Oahu has closed dine-in sections of restaurants, and Kauai has implemented a curfew. Government is doing everything they can to encourage people to make the choice to stay at home . . . because being cooped up at home feels much better when it's YOUR choice rather than something the government has forced upon you.
Unfortunately, there are still so many hard-headed residents and visitors at the malls, beaches, etc. The most stringent measures just kicked in today (Friday) and this weekend will be a litmus test. If large groups are still gathering in public places this weekend, I would expect a shelter-in-place order to come down early next week.