I'm a current MPH student at Brown, and your thoughts on Brown aren't completely true! Brown has a full school of public health. It has its own building between the main campus and the medical school and a slew of undergrad, masters, and PhD programs. In general, I agree with your statement that Brown isn't known for public health because its social science programs have been historically stronger, but the school is new-ish and has a growing reputation in the field. It is certainly a smaller program than Columbia and BU, but that's not necessarily a bad thing. I'm happy to answer any other questions you may have about Brown here or in PM!
I did not apply to Columbia last year because NYC is not for me, but I wouldn't say it's dangerous to live there at all. At least not any more so than Boston or Providence. I currently live in Providence, spent >5 years in Boston, and have lots of family in NYC, but obviously my opinion on this is subjective. I would agree that its reputation in most areas of public health is better than BU's, but if you want to work in Boston after graduation, I'd choose BU for sure.
Have you priced out exactly how much each program will cost you and the cost of living in each city? A lot of schools, including Brown and BU, give a lot of merit scholarships that don't really bring down the cost of the degree in comparison to other places with lower tuition (who tend to give fewer scholarships) all that much.
These are 3 great options, and I don't think you really can go wrong! Congrats!