The best airport to use for Verbier is Geneva.  Apart from car rental or private transfer (under 2 hours), the quickest away to get to Verbier from Geneva airport are the direct shuttle buses run by various companies including AlpineExpress and Alpina Transfers. These transfers take the same time as driving but you will need to follow the time of the shared bus. There is also a connecting train service via Martigny and Le Chable. From GVA (station under the airport!) take the train to to Martigny in under two hours.  At Martigny you change to a small train that will take you to le Châble in 30 minutes. At le Châble you have a choice:  either take the yellow post bus up to Verbier (15mins) or the cable car which runs in the winter and summer seasons (5 mins).Total travel time is about 3 hours, slightly more if you are unlucky with connections.

If you fly to Zürich you must take a train to Lausanne, and then change for Martigny.  Total journey time Zürich/Verbier is around 4 hours.  Basel airport is also an alternative - slightly longer by public transport than Zurich, slightly quicker by road than Zurich. A final alternative is Milan's Malpensa airport.  From there you take a shuttle bus to Gallerate, and get a train to Brig, changing to a train to Martigny.  Total journey time is around 6 hours, but varies as the waiting time in Gallerate my be up to 2 hours.