What if we miss our train connection?

In general, I'd just take the next available train. In the specific case you mention, i would just take the 12:07 RE2 to Appenweier and change to the SWE train to Strasbourg. The conductor will look at the exiting ticket and know the inbound ICE was late. On a regional train, you might not even get checked.

In other cases, if you had a seat reservation and the new train was also an ICE or IC, just stop at the ticket office and get a new seat reservation.

It is also not unheard of that they will hold the connecting train if the the inbound train is a little late, so even if you are running late, assume you will make the connection, and only go to plan B when it is clear you missed it.

In your described case let the train staff do a remark on your ticket or a separate paper that your first train was too late. With this you can use any DB train connection with same or lower classification driving the same route to destination.

Connection classification:
High to low: ICE, IC / EC, RE, RB, S-Bahn.
Example 1: If you booked Regionalbahn (regional train before) you shall not use ICE, IC or EC connection. Of course you can ask the staff of the incoming train - often they are very accommodating.
Example 2: If you booked ICE before you can use any train.

Official statement by Deutsche Bahn in German language.

Use DeepL for translation.

I had that problem a few times and in both cases, I just got on the next train and no one said a word about it. The first train was a regional train and I don't think I was checked, but the second one was a high-speed train and the conductor just nodded and moved on.

Where are you originating, Frankfurt airport (FRA) Fernbahnhof or Frankfurt Hbf (main station)? If, as I suspect, you're flying into FRA on a trans-Atlantic flight, I hope you don't have advance purchase, train specific (Saver or Super-Saver) tickets. There is a bigger chance of missing your first train due to a flight delay than missing your train connection in Karlsruhe. If you miss you first train of a "Saver" ticket due to a late flight arrival, you are SOL.

Assuming your tickets are fully flexible, Flexi- or Flexi-Plus tickets, just wait until the next train(s) to Strasbourg, either another ICE or a TGV directly to Strasbourg, or you can take an ICE to Offenburg and catch a local train from there to Strasbourg.

If you do have Saver/Super-Saver tickets and you manage to make the first train but get to Karlsruhe late, your missing the connecting train is the fault of the Bahn, and they will make it up to you. If you have time, go to the DB desk in Karlsruhe and they will re-issue your ticket for the next train to Strasbourg, but it looks like if there is a train to Strasbourg leaving immediately, just get on it.

I did something similar in 2008, when I was coming from the Harz on a regional train to Göttingen with what was then the equivalent of a Saver ticket. The regional train unexpectedly stayed in a small station for half an hour, and I missed my Göttingen connection, an IC to Karlsruhe. I went to the desk, and the clerk checked the schedule and found that my train had been delayed. He wrote the number of the next long-distance train on my ticket. It was an ICE, and I actually got to Karlsruhe before the IC I was supposed to take did.

If you are on a Flexi- or Flexi-Plus ticket, it is valid on any equal or lessor train to the destination that day. Just get on the next train. Note: if your scheduled train is an IC, and the next train is an ICE, you should probably get your ticket modified at the DB counter.

The tickets I purchased are the Sparpreis Europa. It allows cancellation before the first day of validity for a fee with a voucher. We arrive in FRA and stay one night, then take the train the next morning.

In the originally described case that the follow-up train is missed due to too late first train there is no re-issueing of ticket necessary; just a written confirmation by staff of first train. The booked fare is not relevant, only the booked train category.
See official statement of DB in #2.

"there is no re-issueing of ticket necessary; just a written confirmation by staff of first train."

Not necessary, that's just a suggestion. I'd never queue up and then risk to miss the next train.

The tickets I purchased are the Sparpreis Europa. ... We arrive in FRA and stay one night, then take the train the next morning.

Very good. It wasn't obvious from your original post that you had built in a day's buffer against a late arrival. It's never a good idea to plan on using a non- or limited-refundable ticket right after arrival on a trans-Atlantic flight. The best plan I know of, if you want to leave on a long-distance train immediately upon arrival, is to book a Rail&Fly package through your airline, if offered.

I'd never queue up and then risk to miss the next train

In my case, there was no queue at the DB counter. Just walked up and showed the clerk my ticket. He wrote the number for the next long distance train, which happened to be an ICE, even though my original ticket had been on an IC, which was slower.

I'm not sure what would have happened if I had just jumped on the ICE without being re-ticketed.

Ironically, I would have booked a ticket using the ICE, but at the time I booked, the lower price-tiered tickets for the ICE were all sold, and it was considerably more expensive than the IC, which only arrived in Karlsruhe a few minutes later than the ICE.

The ICE is a "higher" category train than the IC I had booked. I'm not sure what would have happened if I had just jumped on the ICE without being re-ticketed for the higher category train.

The ICE is a "higher" category train than the IC I had booked. I'm not sure what would have happened if I had just jumped on the ICE without being re-ticketed for the higher category train.

Nothing would have happened, because the change IC/EC > ICE is freely possible
in case of delays. It's only the change from local to long-distance trains that is
subject to restrictions. But since the OP will transfer from an ICE, this is
a rather academic discussion - "downgrading" is of course always possible.

In my case, there was no queue at the DB counter.

Congratulations. But since the pandemic, it is unfortunately not uncommon that
DB's travel centers are understaffed.

Nothing would have happened, because the change IC/EC > ICE is freely possible
in case of delays.

I doubt this because when booking IC/EC and ICE have different prices for same route and destination, e. g. Berlin - Hamburg.

Maybe you want to collect some information first? After all, we are not dealing with arcane science here:

Bei einer zu erwartenden Verspätung ab 20 Minuten am Zielbahnhof [...] können
Sie mit Fahrkarten der Produktklassen ICE und IC/EC die Reise [...] mit geänderter Streckenführung und/oder Produktklasse bei nächster Gelegenheit oder zu einem späteren Zeitpunkt ohne zusätzliches Entgelt fortsetzen. ->

"If a delay of 20 minutes or more is expected at your destination [...] tickets of the ICE and IC/EC product classes enable you to continue your journey [...] with a changed routing and/or product class at the next opportunity or at a later time without additional charge."

No need to play Cassandra, then.