Salzburg or Hallstatt in December? It's a difficult choice and you'll have to play it by ear. Salzburg has a Christkindl Market and the whole town is up to its festive best. Also, tourist crowds are thinned out considerably, so that's a plus. At the same time, you won't be able to see a number of outdoor attractions like the Mirabell Gardens or Hellbrunn. Also, if there has been a lot of Snowfall, Salzburg streets can be slushy.
Hallstatt is much more scenic than Salzburg in the winter, but typically colder. The town lies in the shade for most of the time, and I don't think it gets any sun at all in December. Tourist crowds are practically gone, and you'll have much of the town to yourself. The lake rarely freezes solid, and it hasn't done so in recent years. The ferry from the train station still crosses in the winter, if the lake is open. I know of people who have walked across the ice when it wasn't, but I wouldn't recommend it. You can also get there by bus.
So play it by ear. If you have time, give Salzburg a quick once over to see if you like the conditions. If you don't, go to Hallstatt.