From searching online, I see that the salaries for math adjunct professors are about 36k. That's insanely low for New York City standards. An undergraduate business school student that graduates from a decent program will make 3x more money in his first year of work than a Math Phd Adjunct Prof. My question is: What else do these adjunct professors do to make ends meet? Do they typically take on another job? A summer job? Work night shifts at a bar?
30% of households in Manhattan have an income lower than 36k, according to census data; 36k is actually above the average income for an African-American or Hispanic household in Manhattan. You're right that it's hard to get by on that amount of money, but plenty of people manage it.
EDIT: A good point in the comment below: typically graduate stipends are well below this number (31k is the highest I have ever heard for a graduate stipend in math; I'm sure the CUNY stipends are much lower). Of course, most people don't want to live like a grad student forever, but it's not as though it's impossible to survive.
I worked as a math adjunct in NYC for most of the prior decade, making about that much income, no other job, and expenses of about $25K/year. I have a partner who splits expenses with me 50/50 (prior to moving in together we had other roommates who did the same). We live in a nice neighborhood in Brooklyn, but it's very far from Manhattan; we live frugally but comfortably (no kids).
In discussions at school (we've had ongoing contract negotiations for a few years now), the impression that most other staff has is that adjuncts are either public high-school teachers moonlighting for supplemental cash, or spouses of other full-time employed people doing the same. I'm not sure I agree with that characterization (it certainly wasn't true for me).
One other thing about teaching in NYC (at least at CUNY): unlike almost any other location, adjuncts here get full health benefits (up until last year via the PSC-CUNY union; now from CUNY directly). So that's actually an enormous leg up over the same job in other locations. Also: You can easily get by without a car here. All things considered, I was able to save more money adjuncting in NYC than I could previously in Boston.
Someone taking such a job almost always is not depending on it as the sole source of income. They may have a spouse who has a more high-paying job in or near New York City. Alternatively, they may be independently wealthy or retired, or at least have enough money to make ends meet for a year or two while they are forced to stay in the city for some reason.
Certainly, someone who takes such a job is doing it because, for whatever reason, they want to be teaching instead of doing something making more money, and they are lucky enough to have another source of income so that they can teach.