The closest airport is the Geneva Airport (GVA). Here are the itineraries from there.
By Train (around 3h)
The closest train station is Martigny. From Martigny you need to change for a small red train (named the St Bernard Express) who will drive you to Le Chable. From there you can take the ski lift to Verbier or the postal bus.
By Car (160km)
From the Geneva Airport. Take the A9 to Martigny and then follow Grand St Bernard road. When you arrive in Sembrancher you need to turn left and follow the signs Verbier. Since you arrive in Verbier you need to turn left and follow the signs Les Elfes.
It is required to have a car that is equipped with snow tires and if possible chains, if visiting in the winter.