Zurich to Brussels by rail

How does it work? You buy tickets from Switzerland to Brussels.

The Swiss Pass will cover the portion of the journey - if the pass is still active - within Switzerland.

The price difference on such a long journey is unlikely to be significant.

If you are in, say, Zurich, use the pass between Zurich and Basel and buy the ticket from Basel (or wherever which route you choose crosses the border) to Brussels.

If you are dealing with a human, or using the SBB website indicate that you will hold a valid Swiss Pass and the computer will take that into account.

If you know the date, you can buy the Basel Bad to Brussels portion on line 92 days out on www.bahn.com.
The train at 11:00 or so has a single change at Frankfurt Hbf. This is not the fastest route, but most sites will route you through Paris where you will need to transfer from Gare de Lyon to Gare du Nord train stations on the RER D line. Might run a bit more money than sticking to DB. The train to Frankfurt actually starts in Zurich at 10:00, stopping at Basel SBB then Basel Bad. You could ride on your Swiss Pass as far as Basel Bad, then your DB ticket would kick in. Right now I am seeing Sparpreis fares of 89 euro pp for June.

There is actually a direct train from Basel to Brussels.
As Sam says, the quickest route is TGV Zürich --> Basel --> Paris, cross Paris, TGV/Thalys Paris to Brussels. You can book this through www.sbb.ch but not through www.bahn.de as it does not go through Germany.
But, there are 2 direct trains per day from Basel to Brussels, via Strasbourg and Luxembourg:

Basel SBB 13:16, train EC 90, arrive Bruxelles-Midi 20:27
Basel SBB 16:21, train EC 96, arrive Bruxelles-Midi 23:27

Coming from Zürich you have to change at Basel, but you avoid crossing Paris. You can book this through www.sbb.ch or French or Belgian Railways

Nice find Chris. Its not showing up at DB. Its marked EC Vauban. Is this a private railway that doesn't show up in DB data base? Standard fare of 101 chf is decent too.

Sam, that's it, EC Vauban. It is a perfect normal EC train that has been running for years. But as it is slower than the TGV route via Paris it only shows up if you do a search for intermediate destinations, "via Luxembourg" for example. Dumb computers
If you look at this mega PDF (which is the yellow departures poster for Basel SBB) you can see the Vauban at 13:16 and the Iris at 16:21.
It is an old fashioned engine-and-coaches train. It used to come from Zürich, but I don't think it does an longer.

Edit: It is so venerable, it even has a Wikipedia page devoted to the train: Vauban and Iris.

Thanks for all of the info—very helpful and much appreciated!