Going to Hunter for Psychology is a bit of a mix. Expect adjuncts (from Pace, from private practice, some still getting their PhD.) Almost all of my psych. professors at Hunter were students getting doctorates. That's a reality, unless you specifically schedule for tenured doctors.
With that in mind: these are some amazing professors. There's a secret going around that research universities are full of professors that just don't care. Hunter's professors really care, I have to say. They may be about your age, or not speak English entirely fluently, or might even be learning some of the material as they go, but they have a special sort of energy and excitement about it that a 60-year-old, miserable old tenured monster of a professor won't have.
So, pros? Essentially free (no world-crushing debt), full access to professors and labs, no huge egos. Cons.. No prestigious name, probably won't be as demanding as private schools, though that's up to you. You'll publish your own papers, and do your own research, and that's what counts for the PhD.