This summer my wife & I had a wonderful holiday in Switzerland. We flew in/out of Geneva and after much deliberation decided to rent a car at the airport instead of taking the train. Because the airport straddles Switzerland and France, and thus there are car rental agencies on both sides, we searched and searched in vain for current, accurate information on how this worked. We found a few somewhat helpful tips, but ultimately we had to figure it out for ourselves once we arrived. For those in a similar situation, here is current advice as of summer 2011.
At the time of our booking (late June 2011) we found the best rates on the French side (in Euros) versus the Swiss size (Francs). We were concerned about drop charges, differing policies given these were the same agencies but in different countries, insurance, etc. And also how were we to get from the Swiss (where our international flight arrived) side to the French side (where we rented our car, from Hertz).
Hertz offered substantially different rates at the Swiss counter (much higher, factoring in the USD conversion) than at the French counter.
To get from the Swiss side of the airport to the French side is extremely simple: follow the signs, go through a couple of corridors and voila, you are there. No guards, passport checks, nothing. And if you go back to the Swiss side and then want to go back to the French side, it's similarly extremely simple and easy too.
Renting the car on the French side was a breeze and we saved hundreds of dollars versus the train pass and also renting on the Swiss side. Hertz was easy to work with, no hassle/pressure to buy insurance (which we didn't need and verified w/our insurance carrier in the USA and credit card company too) and we were off to the car in about 10 minutes.
Best part was they gave us printed directions, in color, with photos of exactly how to get out of the airport to Switzerland and back - never had any car rental company anywhere in the world do that!
So if you are contemplating a similar Swiss vs French side rental at the GVA airport, relax and don't worry, it's easy and we wished we had known this before we left.