Hi, just assume that there are two airports as far as logistics and law is concerned. They may all be one building but they straddle the border. You are going to land from America into the Swiss (main) side of Geneva airport. For simplicity’s sake just accept there’s no way from the Swiss side to the French side “within” the airport. You’re going to arrive in Switzerland, go through Swiss customs and passport control and emerge onto the sidewalk in Switzerland. From the Swiss side ( which is the major side) of the airport you would naturally go to the Swiss rental car area.

There isn’t anything to stop you hiring a car from the French side ( it’s usually significantly cheaper) but understand the logistics: you will need to exit the airport, find transportation to leave Switzerland and enter France (about a ten minute circular drive) and then make your way to the airport from the other side. It can be tricky to do because sometimes taxis dont want to exit the country: you should find one to take you, but it’s a slight risk.

Important: bear the same process in mind for returning the car. You’re going to have to drop it off “in France”, leave the airport and take a cab through the border, back to the Swiss side of the airport to get your flight home. There’s no way to drop your car off and then wander through to the Swiss side. And if you return your car to the Swiss side expect a massive penalty for dropping it off in the wrong country! People get confused by this all the Time because it’s all one building….

Hope that’s helpful