In my opinion, you are dealing with the "restrictiveness" of clauses or bound/free modifiers.
This sentence can also be written as:
... people who/that studied in universities...
... people who/that were studying in universities...
The participle modifies the people and restricts it to a certain subgroup. So, with that in mind, that/who (relative pronouns) can be omitted IF THERE IS NO AMBIGUITY. That is a case by case situation.
Now, if we omit the relative pronouns from the sentences:
"There was a sharp increase in the number of people studied in universities from 1997 to 2007."
"There was a sharp increase in the number of people studying in universities from 1997 to 2007."
The first sentence implies that the increase occurred in people studied (as in test subjects in academic experiments), while the 2nd one creates no ambiguity; the modifier modifies the correct subject: people.
So, I would tell your student that if he/she is going to omit the relative pronoun in restrictive clauses, he/she should check the implication of the new sentence and ensure there is no ambiguity.