Absolutely the best way of doing this is by cab, but if that isn't viable then there are good bus services that are pretty cheap.
The Bronx is nowhere near as bad as it used to be. I would recommend, though, that he doesn't stroll around on his own - especially at night. But that applies pretty much anywhere nowadays.
I'd consider his vulnerable time will be during travel JFK (I assume) and where he's staying, but daytime bus transportation is OK in NYC. Apart from that, he'll be fine as he's with locals.
I understand your concerns! Last year, my sister and BIL unwittingly booked an AirBNB in a rough part of NYC, and I voiced some concerns. Turns out the extremely dodgy locals liked them, looked after them, and gave them the time of their lives.