The nearest airport is Aspen/Pitkin County (ASE), about 20 km away, with direct flights from Denver, Dallas, Chicago, and several other US cities in winter. Aspen is a small airport with notoriously weather-affected operations — flights are routinely diverted in snow or fog, so build slack into your schedule.
Many visitors fly into Denver International (DEN) instead and either drive (about 3.5 hours via I-70 and Glenwood Canyon) or take a shared shuttle service. The drive is straightforward in summer; in winter, snow chains or 4WD with snow tires are sometimes required on Independence Pass routes (which is in any case closed November to May — the Glenwood Canyon route is the only winter option).
A free local bus (RFTA) runs frequently between Snowmass Village and Aspen.