Ranking =/= reputable.
Like
@MedicineN'Jazz alluded to, Rutgers is not very well respected in the NYC/NJ area, and I'd go so far to say as that reputation holds true across the mid-Atlantic & New England regions. This isn't to say it's a bad school, poorly ranked school, or that the public health education isn't good-- I don't actually have any evidence that any of those things are true. What I do know, however, is that Rutgers as a university is widely regarded as a bit of a joke in the region, although it may not necessarily deserve this reputation.
Outside the region, however, it's reputation is pretty good, on par with most top state schools. I grew up in NJ and was subsequently
floored during my time living in Ohio when people would talk about Rutgers academics as on par with Ohio State, Michigan, etc. after hearing most of my peers/town compare it unfavorably to our local community college my entire life. As another example of this, I knew someone with a BS from Rutgers with an insanely high GPA and impressive resume who was subsequently rejected from all grad programs in the tri-state area and accepted to all grad programs (many of which were more competitive) outside the region. In certain fields, Rutgers' reputation is excellent and many programs are highly ranked, though it's low on the public health US News rankings.
Mt. Sinai, on the other hand, is part of a major, old, and very respected hospital system. Icahn has a higher ranking and much better reputation than NJMS and RWJMS, and often times people (wrongly) conflate med school reputation with public health reputation. In general, the Mt. Sinai/Icahn name will get you much farther in the NYC area than the Rutgers name. But all this said, choosing a public health program should be more about fit than ranking and reputation. I hope
@phm1234 is looking at course lists, faculty research, program size, etc. in making their decision as well, as I'm sure you did.