Don’t big cranes have big counterweights?
Of course. Which is why, when they pass their center of gravity, they can crush apartment buildings.
Everytime I walk by one of those things, I marvel at them; and of course I wonder what it would be like to be in one and feeling that awful feeling...
Not really, with this kind of crane, assuming it's the kind most commonly used for Manhattan high-rises. A small counterweight way up at the top, on the horizontal part, but nothing that would have any significant effect on stabilizing the whole crane. These are vertical cranes, that are constructed floor by floor as a building gets taller, immediately adjacent to the wall of the building. The vertical portion is normally secured to the building, and only the small horizontal portion is moveable and counterweighted. If this crane's vertical support wasn't properly stabilized, no counterweight could have saved it. A picture of one (on a much taller building) is at this link: http://www.dreamstime.com/highriseconstructioncrane-image207562
I watched one of these things grow out the window of my then office building at Lex and 57th about 10 years ago. Pretty cool. There's a mechanism that lifts the existing sections, opening up a space near the bottom to insert new sections one at a time. I wouldn't be surprised if such an insertion was in progress when this one went over.