From personal experience, I would not rule out any of these areas based on the "danger" of them which is largely overblown, especially if you have been reading about this danger online.
I have personal experience with NYU, Columbia, Chicago, and to a lesser extent Upenn. For NYU, Columbia, Chicago, it is quite safe most times of day except perhaps very late at night (2-4am for example, but even then you are probably fine). To my knowledge, all of these campuses have safe ride programs where you can get a campus security shuttle or escort from anywhere on campus to another university property and/or your apartment depending on the set up.
That being said, common sense is your best asset and living close to campus is probably safest (due to the density of students and campus police). At any time, don't walk about with your iPhone, iPad, etc out highly exposed where someone can snatch it; don't walk around aimlessly or look like a tourist or that you are lost. Late at night (much past midnight) and if you are a girl, it is probably best to walk with someone else, and taking a bike is a great alternative because you are not nearly as easy a target on bike. (For any of these locations, girls seem to be fine walking by themselves at night, and I do not know of anyone personally who has had a dangerous experience walking by themselves).
My advice would be to apply to any and all these programs if they fit your interests, and then if accepted, go to the visit day to get a feel for the campus. Talk to other students there with your concerns and make a choice. Having attended NYU and worked at Columbia, I was initially under the impression that Chicago was far too dangerous. This impression turned out to be vastly incorrect. Chicago/Hyde Park is actually far calmer and quieter than NYU/Columbia and based on the fortification of NYU/Columbia and lack thereof in Chicago, further signals to the comparatively safer feel of that school versus the others. This being said, I never felt unsafe at Columbia/NYU even walking/biking about between 2am-4am.