If it dumps the Pagosa side of the pass can and does close. It's rare that the South Fork side closes. It does happen, but 99.9% of the time, when the pass is closed there is a foot note saying that it is open to the ski area from the South Fork side. The most dangerous avalanches paths that threaten the pass are all on the west side of the pass.
Pagosa is the nicer town to stay in. Better restaurants, bars, hotels, and shops. South Fork slide pretty much guarantees that you'll get to ride the whole time. Just not a lot going on there after you ride.