How happily will taxis cross the triple-point borders around Basel?

I've taken taxis from the Basel EuroAirport (which is in France) from both the French side and the Swiss side of the airport to my home in Germany.

The French taxis drive straight east across France into Germany.

The Swiss taxis go south through Basel, then back north again to Germany, which means a longer and more expensive ride, as if the Swiss taxi rates weren't high enough already. This is because the French and Swiss sides of the airport are separated outside (though you can cross inside the airport), so the Swiss taxis have to follow the customs road to Basel. You can read more about this here: May a cyclist or a pedestrian cross from Switzerland to France near the Basel EuroAirport without going into the airport?

So they will all get you to Germany with no problem, but Swiss taxis will take a longer and pricier route.

However, I would also point out that Swiss public transit is excellent and worth considering. Bus 50 can take you from the EuroAirport to Basel SBB, the main train station and transit hub in Basel. From there you can get basically anywhere by public transit.

Public transit may even be covered in your hotel stay. I don't know about France, but many hotels in Switzerland and Germany include travel cards that provide you with free travel in a large area. Sometimes the hotel or the city you're staying in can give you a temporary pass (which you print) to get from airport to hotel for free. It's worth asking.

To add to the answer of @andra. I looked at the websites of the taxi companies of Basel. While again, there is no explicit mention of it, one of the companies specifically states as their motto:

Der Shuttle Dienst im Dreiländereck Schweiz, Frankreich & Deutschland

This translates to "The shuttle service in the triple-point Switzerland, France and Germany". Furthermore they have an online booking request feature, which seems to accept both German and French places when I tried it and give a valid price. (To be fair, they seem to accept about any place that google maps can route to, so maybe you better send them a request first).

Based on that I would agree with the other answers, that it shouldn't be a problem to get taxis to drive over the border.

On a different note, I strongly recommend you to book a taxi by phone, as just hailing a taxi is not very common in Switzerland, even less so if you are outside of the city centre. Furthermore as you will be going there during the period when Basel World is taking place, it might be quite difficult to get a taxi in the first place. (I guess that is also the reason you can't find a room in Basel).

Disclaimer: I have never took a taxi from France/Germany to/from Basel.

It seems to be possible. On tripadvisor it is mentioned that you should expect a cost of CHF 150. Also the wikitravel entry on Basel mentions a museum in Germany which is reachable by taxi:

Vitra Design Museum, Charles-Eames-Str. 1, Weil am Rhein, Germany, +49 7621 702 3200. No collection, only changing temporary exhibitions - see website [26]. But worth the visit for any self-respecting architecture junkie: During a guided tour of the campus (Tuesday - Sunday, at 12PM and 2PM, 2 hours, call ahead), see Frank Gehry's first European building, and more structures by Zaha Hadid, Tadao Ando, Nicholas Grimshaw and Alvaro Siza. Bring your passport and take bus no. 55 from Claraplatz or Basel's Badischer Bahnhof to the stop "Vitra" in Weil am Rhein (15 - 20 minutes). You can also take the train to the trainstation Weil am Rhein. From there schedule a 15-minute walk to the museum. - Or take a taxi from downtown Basel (15 minutes).

Based on these two internet pages, I assume that taxi's regularly cross borders.

I couldn't figure out with a decent bit of certainty if taxis do indeed cross the Swiss/French border in Basel happily. All the websites point to taxi services to and from the Euro Airport, and these seem to be split between the French and Swiss sides of the arrival exits. You can try calling any of the French taxi services listed to check if they do indeed cross into Switzerland.

Just to add a different option, if taxis won't take you across the border, I'm prepared to bet that you'll find a car-sharing frontalier traveling the Mulhouse-Basel route willing to take you onboard. Provided you are legally allowed to immigrate Switzerland, of course. See here for the results from a sample search on blablacar.

For completeness sake note that whilst there might be no immigration controls, Switzerland does enforce customs checks. The Swiss post on the way to Basel is definitely manned. Switzerland is inside Schengen but outside the Customs Union and therefore performs it's own customs controls at the border. On your way out of Switzerland however the border post is often empty on both sides, especially during peak hour when all the frontaliers go home. This is all based on personal experience. I've been going through the Basel border roughly every two months, in both directions and at all times of the day (including peak hour yesterday evening).