Which (if any) is the "most major" airport / field / strip on Antarctica?
You can find a full list of airports on the continent here, the majority are ice/snow runways with two gravel runways at Rothera Air Facility and TNM. Supported is a loose term down there, however Williams Field which services McMurdo Station is groomed and has fuel available. Phoenix Airfield which also serves McMurdo can accept wheeled landings and seems to be marked in some capacity. Marble Point (a helicopter landing site) is also operated by the US, has fuel available and is operable 24 hours a day in the summer and has weather reporting. 

The full operations of the (US) bases can be found in this document which states 


  Williams Field is serviced by MLS, TACAN, and RNAV (GPS) approaches.
  
  ...
  
  Phoenix Airfield is serviced by TACAN, MLS and RNAV approaches. 


which is a solid set of approaches all things considered. 

At least two fields have runway lighting 


  Airfield lighting systems are established at Phoenix and Williams
  Fields. The NSF Prime Contractor maintains airfield markings and
  lighting systems in accordance with AFMAN 32-1076


McMurdo even offers ATC!

All that considered Williams/Phoenix are your best bets but if you are trying to fly there on your own keep in mind: 


  All navigational and approach aids are intended strictly for use by
  USAP (United States Antarctic Program) approved aircraft only. The use of NAVAIDs by non-USAP aircraft
  are strictly at the pilot’s own risk.


All of the fields are "weather permitting" as are the flights so "open" all year is relative to weather conditions and other considerations.