What could be done to make it easier for a student to obtain a higher degree?

In the United States, open access to higher education is an accepted fact. The idea is that anyone can obtain a degree, as long as they work hard. Is this true? Are there barriers in the way of obtaining a degree for some students that aren’t there for others? What are these barriers? Should higher education cost less, i.e. should the state (the public) pay more of the cost? What could be done to make it easier for a student to obtain a higher degree?