A bit confusing but I'd go for "Yes" in answer to your last question.
I'm about to move over the border to France and will be paid by a swiss company - but of course my tax obligations are with where I live, rather than where I am paid.
If it's just next to the last roundabout before you get to the airport - where Jet Aviation admin buildings and the SWISS building are - then that's the Swiss sector, in Switzerland.
Basel Airport territory is a peculiar thing and all authorities absolutely avoid issuing any official statements about where something is and what laws should apply.
But even if you are working for a Swiss company located in the Swiss sector who deducts Swiss social charges, if you live in Saint Louis, you will submit your income declaration in France and will not be taxed in Switzerland.
(Sorry, don't know how it works in case you are considered an UK resident. )