Yes, people should pay for university, because the more a person invests in something, the harder they are to work at it. When a person pays for something, they care about whether the job is done well. If students have to pay, they will only go to university if they really want to be there.
Personally, I believe that people should pay for their schooling past high school, rather than being granted a free ride. However, they should also apply for as many scholarships and grants (i.e., aid that need not be paid back) as they can, to help alleviate some of the costs. Paying for one's education helps increase one's appreciation for the schooling they are receiving, and can also encourage one to pursue good grades. That being said, if a person has earned grants, scholarships, etc. through merit, that person deserves to receive aid based on all they have done in order to earn it.