There is a shuttle bus from Annemasse - take the train to there first and then the bus.

Otherwise, we have no option [from Evian] but to either go by car, or to take the ferry across to Lausanne [VERY expensive] and then the train to Cornavain/Airport

There is also the bus 61 from Annemasse Gare which you can either change for the tram 12  at the border at Moillesulaz, or carry on down through town to Cornavin with a iimited number of stops which allow you to connect further in Geneva. These tend to be quite busy dependant on the hour.  

http://www.tpg.ch/fr/horaires/rechercher?p_p_id=PlansReseaux_WAR_PlansReseauxportlet&p_p_lifecycle=0&p_p_state=normal&p_p_mode=view&p_p_col_id=column-1&p_p_col_count=2&_PlansReseaux_WAR_PlansReseauxportlet_jspPage=%2Fhtml%2F_thermometre.jsp&_PlansReseaux_WAR_PlansReseauxportlet_req=thermometre&_PlansReseaux_WAR_PlansReseauxportlet_ligne=61

You can use the tpg itinerary search for further details on this http://tpg.hafas.de/bin/tp/query.exe/fn

Thanks Sheila. Do you know if there's a website with details of that bus?

This shows the complete lack of territorial planning from the French authorities.

They have led ced construction all over the place, with no consideration for transport. (Except of course by  car, which leads to endless traffic jams, and plenty of pollution.)