This is not a definitive answer, but based on some examination of the photos in the NYT article you linked, as well as a couple of others, here for instance & there is also at least one YouTube short biography (beware: link is a video).
A significant number of the images have dimensions which indicate either they came from a camera which has a 5/4 ratio or they've all been cropped. I couldn't find anything which looked like uncropped 35mm (3/2). So I'm guessing that, at least early on, he was using either a large-format camera (presumably 5x4), or perhaps a 6x4.5 MF camera. The 1953 Mingus-Haynes-Parker-Monk photo also seems to bear this out: it's very close to 5/4 indeed, and it's pretty grainless, especially given the date: either it came from a 5x4 camera or perhaps from MF I think.
(It is clear from the beginning of the biographical video I linked above that he did use 35mm cameras as well, at least at some point.)
And, again looking at the 1953 photo, there is plenty of depth-of-field: if you look at Monk's head and the bass drum he's in front of they are both adequately in focus. This can't have been done with movements which can only shift the focal plane around, but can't increase depth-of-field for things which appear in the same place on the image. And we can guess at the exposure assuming it's not posed: looking at Monk's fingers there is some motion blur but not a lot, so let's say it's no longer than 1/2s.
I'm not going to guess at the focal length of the lens but it's clearly something approaching normal, so maybe 35-50mm on 35mm or 140-180mm on 5x4.
So, to summarise: this looks to be a picture taken with a medium or large-format camera, and I suspect with a 5x4 camera, with decent depth of field, and a reasonably short exposure, in a club without flash.
Let's assume that he was using film which was rated at ISO 400 or less (I don't know the history of fast films, but I doubt anything practical faster than 400 was available in 1953): TXP say (which I think was still 200 at that point in fact).
What can we say about that? Well at first blush I'm thinking that I must be wrong. I use a 5x4 camera and trying to get that much DoF in a club would be a seriously heroic feat: if this is 5x4 it looks like f/22 or slower. But, well, what could he have done?
- The film could be pushed significantly, and if it is indeed sheet film or MF this will be OK as there's so much to play with. Perhaps it could be pushed to 1600?
- He used lights. He didn't use flash, but nothing says he didn't use lights and in fact the thing you quoted pretty much says that he did, at least early on. There is some other evidence for this in the photo: I think the things that look like bits of bent metal bolted to the wall are lights, but there's no obvious light coming from them: either they're off (why) or they're being drowned by some rather bright light or lights out of frame. My vote is the latter.
So my guess is that, early on (and in this picture I think) his technique was probably being willing to push film, and the development of some lighting system he could drop into a club without having people get cross at him. I would guess that later (when it says he worked by available light only) he started using 35mm and fast lenses, and perhaps still pushing film a bit. It's very possible to take pictures in clubs with ISO 400 film and a f/1.4 lens: I've done this. Looking at other pictures of his he was also willing to print pretty dark, so most of the person is in shadow, which also helps of course.
More importantly than either was being there, of course, and getting to the point where fairly-irascible musicians were willing to let you take their pictures, and then taking good pictures.
Given that cameras will now happily work at ISOs of many thousands, I think most of the technical stuff he did is now pretty obsolete. Being there, taking the right picture, and being willing to print fairly dark so things still look moody, still counts of course.