Both are fine. But can be interpreted slightly differently:
I'm studying Korean at the university: I understand this as - you are studying Korean at University at the moment and possibly for a year or so, but you will eventually stop studying Korean at University.
Vs
I study Korean at the university: I understand this as - you study Korean at University. You have been studying Korean at the university for some time in the past and you will be studying Korean at the university for some time in the future. It makes sense, however, If you said 'I teach Korean at the University' it would make more sense to me because professors stay at one university for many years (10 years plus) whereas, generally, most people only study at certain course at a university for a period of roughly 3 years, and then move on to something else.
I hope this helps,
Tim