Black Forest KONUS Card

AFAIK the KONUS card is valid as far as Basel Badischer Bahnhof, which is the station all trains from Germany stop at first, and is run by DB. It is not valid on transport within Basel.

The only ICE/IC trains to Basel run on the Rhine valley line: Karlsruhe - Offenburg - Freiburg - Basel. These are listed on timetables and on the departure board at the station as "ICE xxx". The are big white trains, you can't miss them. (Click here for photo).
There are also hourly local trains Offenburg - Freiburg - Basel, stopping at all stations and a lot slower.
The other two lines into Basel from Germany (Zell - Lörrach - Basel and Schaffhausen - Waldshut - Basel) are regional trains only, no ICE.

Basel is more expensive than towns in the Black Forest, you are warned. You will need some CHF for Basel. Some palces take Euro, but not at the best rates.
When you arrive at Basel Badischer Bahnhof, get a tram into the city centre (No. 6), Buy a 1 day "Tageskarte" from the ticket machine.

Yes, the KONUS card is valid to Basel's SBB station on the regional trains.

The long-distance ICE and IC trains are different in appearance:

https://www.bahn.com/en/view/trains/long-distance/ice-ice-sprinter.shtml

Regional trains are often red and look something like this:

https://www.youtube.com/watch?v=_DZzXA5zUXc

The train type is also listed on the departure board along with destination, platform, and departure time:
https://bahnfotokiste.startbilder.de/1200/abfahrt-muenchen-hbf-am-29082014-365423.jpg

RB and RE are regional. So are IRE trains, S-trains, M-trains between Munich and Salzburg, and dozens of other designations (like BOB in the photo above.) The ICE, IC and EC trains are off limits.

Basel SBB appears within the dotted perimeter for KONUS use on the KONUS map.

Also... If you know German, you can read the explanation on this page that trains between Basel Bad and Basel SBB are covered by the KONUS card (not so the local Basel trams that run between these stations, however.)

Thanks Chris and Russ, this is very useful information. I'm familiar with Basel and its tram system as we have visited the city several times in the past. I'm trying to avoid any driving in Switzerland, especially in the cities as it is hectic and parking is an issue. Since we are spending a few days in the Black Forest (probably Gengenbach), I thought we might plan a day trip to Basel on the train. However, it now looks as if we also plan to visit another relative in Solothurn so I'm now looking at the possibility of one night in Basel followed by a train trip to Solothurn, probably staying for one night. I'm not positive, but I believe that I would have to get to the SSB station to take the train to Solothurn, probably staying over for the evening.

After Solothurn, we are heading to the Colmar, France area for 4 days. It looks like we will have to take a Swiss train from Solothurn to Basel and then perhaps a TER from Basel to Colmar. Do you have any alternative suggestions?

but I believe that I would have to get to the SSB station to take the train to Solothurn

Yes, trains to Solothurn leave from Basel SBB. Along with trains to all other destinations on the left bank of the Rhine (most of Switzerland and France). Trains to destinations on the right bank of the Rhine (mostly Germany) leave from Basel Badischer Bahnhof. Some trains from Germany call at both stations.

heading to the Colmar, France area for 4 days. It looks like we will have to take a Swiss train from Solothurn to Basel and then perhaps a TER from Basel to Colmar.

Yes, correct. Trains to Colmar leave from an annex containing platforms 30-35. Currently, due to building work, you have to exit the main part of the station and walk along the street for about 200 metres to reach this annex. it is signposted.

You can look up train times on the SBB website: https://www.sbb.ch/en