September: Arrive Zurich, a couple of nights (where?), board the ship in Basel: recommendations?

Day
1. Arrive Zurich. Train to Bern (1N)
2. Train to Luzern (2N)
3. Pilatus excursion (depending on weather)
4. Train to Basel to board ship

All 3 train journeys above are 1 hour to 1 hr 20min.

I guess I would skip Bern and go straight to Luzern for those 2 or 3 days (nights). B&B type establishments are not too common in Switzerland, but hotels that include breakfast are. They are expensive though, as is just about everything in Switzerland.

I agree with Sam, all 3 nights in Luzern. Then use one of the days to take a train trip to Interlaken and back. This goes over the Brünig pass and will give you a (small) fix of scenic mountain train).
The other options are boat trip on the lake, and mountain trip (Pllatus, Rigi or Titlis).
Allow for the weather and be flexible. Only go up a mountain if the top is not in cloud. If you get rain forecast, Use that day for the train trip to Luzern, you will be looking at storms in the mountains from inside a dry train.

Also allow at least ½ day to explore Basel, it is a nice city.

Expect to pay at least 100 Franks per night in a hotel, that is the way it is!

You are all AWSOME!
Thanks ever so much. The comment about 100 Francs: that appears to be about $101 US. So that's not really that much for a hotel room.
If you have any other advice, please send it along.

You are the Best!!!
Thanks!
Corky

If you arrive in Basel you could shoot north to Colmar in Alsace and visit the wine villages just to the north of it, a very pretty area and great for hiking. In terms of rainfall, historically a dryish area for that time of the year.

How long it takes to visit the wine villages?