The closest airports in Italy are Milan and Bergamo Airports. While in Switzerland, the closest airport is Zurich Airport.
Milan offers easy connections thanks to the shuttle service between Malpensa and Linate Airports and the Central Station, from which many direct trains depart daily for Valtellina.
From Zurich, instead, you can reach Livigno by train through St. Moritz Station and then by bus or by taxi.