I was in Basel last month and it’s super easy to get to from the Zurich airport. The train station is right in the airport complex so easy to get to. There are ticket machines where you can buy your tickets and train employees nearby if you need assistance. Some of the train routes will require you to change trains at the Zurich main train station or there are some that go direct to Basel with no change. I suggest you download the SBB app on your phone. That way, you can see all the different routes and which ones require a change.
There are multiple train stations in Basel. You will want Basel SBB. Directly in front of the train station is where you will find buses and trams that you can use to get to the old town.
If you don’t have to be in the old town, we stayed at a great hotel right across from the train station—Hotel Schweizerhof Basel.
https://www.schweizerhof-basel.ch/en/
The staff was so helpful and the breakfast was fabulous. The hotel has an elevator. We loved the location because it was so easy to get to. And the trams are right outside so we could easily get to the old town in a matter of minutes. The hotel provides a transportation card so there was no need to buy tram tickets.
I looked on-line and it looks like the Basel Christmas markets are Nov. 25 – Dec. 23rd. Basel is super close to Freiburg, Germany and it looks like that market starts a little earlier on Nov. 18th. You are also close to Colmar and Strasbourg, France but it looks like those markets open later on Nov. 26th.