Galloglaigh is right. It is an easy journey from the station at Geneva Airport to Martigny and on to Orsieres. The bus from there leaves only twice a day as far as I can recall so you will probably have to stay somewhere the night unless you are lucky with your flight times.If you want to work out where to get to from anywhere, try Rome2Rio.
Their answer to the question is here
You go by train to Martigny or more likely Orsieres and take the Postbus
If you decide to stop at Martigny, the Tourist office opposite McDonalds is very helpful and can arrange a taxi if you don't want to wait for the bus. Warning though, taxis and everything in Switzerland is $$$$ but it may be better than hanging around.
Hi I think the best way is by train easy and quick about £15 .We walked from Lausanne to St Bernards pass last year. It is a really nice walk along the lake .All very expensive.All hotel nothing basic and not like the Spanish camino .Hi all,
Just started planning my August 2020 pilgrimage from Grand St Bernard Pass to Rome.
I will be flying into Geneva.
What is the best way to get to Grand St Bernard Pass using public transport and perhaps approximate costs?
Looking forward to your suggestions