Now if you're talking about raising a family on a single income of that much, that's a different story...
So the long and short of it is, $125k, even before tax, is far more than you need to live in SF for a single person. That isn't to say it's wonderful to have to spend so much on rent, but it's beyond hyperbole to say you need $125k+ to survive.
It is certainly "doable" to raise a family on $125,000 a year or less in NYC as evidenced by the fact that millions of people are doing so.
Sure, living in NYC for less than $125k is doable under the following circumstances:
1. You have a rent controlled apartment.
2. You are on welfare or other government assistance programs (which often come with public housing).
3. You have other income (like a trust fund).
4. You live in New Jersey, Long Island or Yonkers and commute an hour each way.
5. You pay the high rent anyway, build no savings because you're spending $2,000/mo on a studio you split with a roommate, regret living in NYC - and move away a few years later.
At least among my friends, NYC is a revolving door of "Yeah, New York is awesome!" and "New York is awesome but I'm a successful 30-year-old professional can't afford more than a studio" to "Fuck this place, I'm moving to Austin."
The only reason there are any poor people left in Manhattan is because the city essentially subsidizes them living there. I have nothing against that, but Manhattan real estate is unaffordable for mere mortals otherwise.
One can live in e.g. Kew Gardens, have a 35 minute commute to midtown, and pay $2000 / month for a spacious two bedroom shared with spouse and child. No trust fund needed.
If your friends prefer Austin to Queens, more power to them. But they ought not to go around saying it is impossible to live/raise a family in NYC on less than $125k when it isn't true.