The closest airport is North Eleuthera (ELH for commercial, MYEH for charter flights). From there it’s a five-minute taxi ride to Three-Island dock to take the ten-minute boat ride to Harbour Island.
There are direct flights from Nassau (Pineapple Air and Bahamas Air), Miami (American Eagle), Fort Lauderdale (Silverjet and TIA), Atlanta (Delta), and Charlotte (American Eagle), depending on the time of year.
If you are coming from Europe, it’s best to fly to Nassau. There is a direct British Airways flight from London Heathrow, usually four days a week (typically Mondays, Thursdays, Friday and Sundays) and twice weekly Virgin atlantic flights from Heathrow (Monday and Friday during the Summer, Wednesday and Saturday during the Winter).
Alternatively, transferring via Miami, Fort Lauderdale, or Atlanta are good options, but they usually involve an overnight stay in the U.S., unless you charter a plane to North Eleuthera.