Colleges are expecting what could be the largest freshman class ever this fall at a moment of extraordinary turmoil, as campuses face financial pressures from the federal government and political conflict over diversity and other cultural issues.

Admissions processes, upended by the Supreme Court decision to ban affirmative action, have been revamped. Budget pressures and worries about financial aid and tuition loom for colleges and families alike. Campuses have been grappling with protests…