Many possible itineraries - depends mainly on what you are looking for (and what you call "close by"). Alpine scenery? Other Christmas markets?
And yes, using trains in Switzerland is expensive. You can easily get an idea with www.sbb.ch timetable - from Basel to some tourist places such as Lucerne, Montreux, Berne ... . Once you get the possible itineraries, open one and check price information (full fare). At the moment, timetable info is limited to December 14, 2013 - prices will be comparable before and after that date. Works for intineraries in Switzerland.
I would consider visiting other Christmas markets - but in nearby France (Alsace in fact) - Colmar or Strasbourg. And buy tickets directly from French national railway (SNCF). Price information from their site too.
But Basel has lots to offer - definitely much more than just for a day' Christmas markets.