There are a growing number of mentally-ill students on campuses today, thanks in part to the success of psychotropic drugs, which allow more troubled students to get to college and to do well there.
There's no need to hyphenate mentally ill. Mentally is an adverb that modifies ill. Use hyphens when stringing together mutiple adjectives that modify a noun, because they make a sentence's meaning more clear.
For example: The high-intensity university life can be hard on students who are mentally ill.
One more note: It should be "There is a growing number," not "There are...." Number is the subject, and verbs must match the subject. Single subject, single verb conjugation, and so on.
The writer was confused here no doubt because of the prepositional phrase "of mentally ill students." When determining whether you need a singular or plural verb, look to the subject. You can ignore the prepositional phrase completely.