I learnt on a combination of electric guitar and nylon-string guitar.

Electric guitar : I tried very thin strings, size 08 which helped with lowering the tension of the string, so even though they're thinner, they cut into my fingers less.

Nylon string guitars : The strings are much thicker and don't hurt as much. I de-tuned it by quite a lot so that I could still bend the strings (I was learning from recordings of Mr Hendrix). This also lowered the tension, so has a double effect. It might also work on a steel string guitar.

If your students are playing acoustic steel-string guitar, I think thet's probably the most abrasive combination on the fingers as the strings are usually thicker & so quite tight. Playing a nylon string now and then would still train for the chord shapes etc but hurt less.

There's also the act of running your fingers up the string while changing chords, which can end up using the wound strings as a file on your finger. You might be able to get around this with round-wound strings, although I'm not sure whether these are available for acoustic guitars.

So in a nutshell:

* Try de-tuning the guitar - less tension = less cutting on the fingers. Not too much though as it's easy to squeeze the strings out of tune.
* Try lighter gauge strings
* Try a nylon string guitar
* If possible try round wound strings

I think others have answered wisely too, especially "stop before it hurts".