Ordinarily any taxi from any of the three countries will cross to either of the other two with no problem.
Be cautious, though, if you're going to or from the Basel EuroAirport. The airport has a French side and a Swiss side (notwithstanding that the whole thing is ostensibly in French territory). There's a "customs road" that runs between Basel and the airport with no entry or exit except at each end.
Thus if you exit the airport on the French side, your taxi can drive straight east across France into Germany.
But if you take a taxi from the Swiss side intending to go to Germany, your taxi will head south on the customs road to Basel, then back up to Germany, meaning a longer and more expensive ride (as if the Swiss taxi rates weren't high enough already).
You can read more about this dual-airport, separated-road situation here: May a cyclist or a pedestrian cross from Switzerland to France near the Basel EuroAirport without going into the airport?
I would 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.