Take the train or bus #5 or #10 from the airport (Cointrin) to the train station (Cornavin).
When you get out of the main entrance of the airport (street level), the #10 bus stop is right in front, towards the left side of the airport facade (with your back to the building). The #5 bus stop is just 5 meters behind (just turn left) the #10 bus stop.
The train station is also very close, just a few steps from both bus stops. It takes 5-10 mins by train and 30 mins by bus to get to Cornavin.
At the Cornavin train station, take tram #13, direction Palettes, and stop at Carouge (the trip is around 18 minutes).
To find the tram stop for #13: Go out through the station's main entrance (not the side or rear doors), then cross the street up to the tram stop sheds in the middle. Tram #13 that goes to Carouge heads to your right when your back is turned to the train station.
The Battelle campus is just off the Carouge crossroads; from the main gate (marked "Centre de recherche de Genve), walk up to Building A (the 1st building on your right). The AI Lab is on the second floor, the wing to your right when you leave the lift.
This map of Geneva shows the three landmarks -- the Cointrin Airport, the Cornavin train station and Carouge---in bright green circles. It also shows the itineraries of the different bus and tram lines. But if you want the gory details, you might try the web page of the Geneva Public Transport System (in French only).