Crime is present on the campus of Seattle University, but not to any significant extent. There have been occasional muggings and incidences of theft (don’t leave your dorm door unlocked or something sitting in a lounge), but generally the campus itself is safe. Public Safety actively patrols the campus, and has introduced a network of emergency phones that enable students to rapidly report when they have been a victim of a crime or might not feel safe at that moment. And with the wireless warning system, if a major security event were to occur, notification can occur rapidly.
The neighborhood off campus is a bit sketchier. There are places you would not want to walk through alone at night, and there has been a rise in incidents of anti-gay violence in Capitol Hill in recent months. Gang-related crimes have also been on the rise in Seattle, and the handling of them has been a source of criticism toward the mayor. However, it is certainly not an epidemic, and should not prevent you from coming to Seattle U.
For those who are going off campus Public Safety offers the Nighthawk Shuttle program and the 24-hour two-block foot escorts, and they are not that restrictive about who goes into a dorm. Though you need a campus ID card to get in and use the elevator, most students will readily card you in and up. The RAs are a diverse lot, with some enforcing every campus regulation all the time and many others willing to ignore certain things like underage drinking, as long as those who do so are safe on campus. The concern all around is for safety, and if the RAs or Public Safety catch you doing something dangerous or stupid they will report you; otherwise, they can let certain things slide.