The French AutoRoute Blanche (A40) ends a few kilometers from St. Gervais where it begins the climb up to the foot of Mt. Blanc and the tunnel that leads to Italy.
The nearest major airport is in Geneva (GVA)  about 45 minutes away by car and is serviced by most major European airlines. Only Borini Bus run a direct service from from Geneva airport to St. Gervais priced at 42 euros (don't be worried by website requests to book your coach ticket in advance, you can easily get bus tickets in the arrivals hall.)
La Fayet, a few kilometers away, is also served by coaches with a single journey costing 19 euros with Ouibus  at the time of writing, and then either a 10 euro taxi or another Ouibus (often timed to leave literally five minutes after the first arrives) for an extra 6 euros. Ouibus will get you all the way there in eighty minutes.