I'm not surprised it's not possible to cross between Switzerland and France in a car around the Euroairport (though I'm also not sure why it's like this)
Route douanière is technically on the French territory, but you are in Switzerland with respect to immigration (which is less of a problem after Swiss accession to Schengen) and customs (which is very important, particularly for the companies dealing with physical goods in Swiss sector of the airport). This is why there are fences and barriers surrounding the road.
The road is first and foremost the treaty-required customs-free Swiss service road for the Swiss sector of the airport. The idea is you should only use the road for the airport (or airport-related businesses operating in Swiss customs territory) and that's why France agreed to "cede" the territory.
Technically they could probably open crossings along it (I didn't read the bilateral agreement in detail), but there are few incentives.
From Google streetview it doesn't appear to be any obvious places where you can cross. Other than customs reasons, airports and many airport-related businesses also have other requirements for aviation security reasons, so a crossing point is probably unlikely.
For the legal provisions:
Treaty between Switzerland and France on the construction and operation of Basel-Mulhouse Airport (1949)
Art. 7 Route douanière
- L'aéroport sera relié directement à la frontière franco-suisse par une route
affectée à son trafic. L'aéroport et la route seront séparés par une clôture du reste du territoire douanier français. Sous réserve des dispositions qui seront éventuellement
arrêtées d'un commun accord en vue de son utilisation pour le trafic général, cette route
fera partie du secteur affecté aux services suisses conformément aux articles 2 et 8.
Art. 7 Route douanière
- The airport shall be connected directly to the French-Swiss border by a road dedicated to its traffic. The airport and the road shall be separated by a fence from the rest of the French customs territory. Subject to any possible provisions mutually agreed for its use for general traffic, this road shall form part of the sector assigned to Swiss services in accordance with Articles 2 and 8.
As far as I could find, no agreement on its use for general traffic was made so it remains a dedicated service road. It doesn't in principle prohibit pedestrian crossings for the fences though, now Switzerland is in Schengen.