'See you in the next course' X 'See you on the next course': Qual é o correto?
As duas opções são corretas. Tanto 'in the next course' quanto 'on the next course' são utilizadas por falantes nativos.
Both can be used, as Donay said, but there CAN be a slight difference, which is not that important, but worth pointing out nonetheless, and I emphasized "can" meaning it's not always the case and they can indeed mean the very same thing, but if there was a difference it would probably go as follows: When you say "in the next course", in the context at hand you are referring to the period in time you will be seeing that person. You'll be seeing them in the time span in which the course will take place, but not necessarily always at the exact moment they're having classes. This applies especially if that person is someone with whom you don't stay in touch that much outside the classroom. Whereas with "on the next course" you are emphasizing and specifying that you will be seeing them as an actual course partner, in the classroom, while having classes.