as said, you are very unlikely to find a US flight to Basel - Frankfurt or Zurich are the obvious airports for US flights, you could check Stuttgart but I wouldn't build your hopes up. You might find a connecting flight to Basel via London Heathrow or Amsterdam which is cheap enough to justify the hassle - and if it was via Amsterdam you would do the Schengen entry process there.
The big question here is the planned date of travel because the german railway which sort of connects the two ends of the walk is closed for a massive rebuild until the middle of 2027, with a batch of replacement buses filling in gaps. This is not a total disaster because with a little bit of sideways thinking you can actually avoid it.
I'm assuming that you can read a map if you intend to do this. Look at Stühlingen, now look 2 miles east, see the Swiss village of Schleitheim? That has a city bus to Schaffhausen railway station every 30 minutes (hourly on Sundays), all have train connections to Zurich Airport, often direct. The journey between Schleitheim and the airport can be less than 90 minutes. Because of the 2 mile gap this gets missed by the usual journey planning sites. And if you intend to walk the trail I take it you can manage this 30-40 minute extension. In the SBB journey planner use locations Schleitheim, Bahnhofstrasse and Zürich Flughafen - ticketing is integrated so you can buy one ticket for the whole journey, even from the bus driver.
For the Wehr end it's almost as easy and in this case the SBB journey planner gets it right (DB get it wrong!). Every hour direct train from Zurich Airport to Basel SBB has connection to Schopfheim with bus to Wehr (the Basel-Wehr section is the same as if coming from Basel airport).