I can't speak for others, but I was deliberately going for a Sixties spy-fi feel in parts of my EUGENICS WARS books, to the extent that I often had the soundtrack to "Our Man Flint" playing in the background while writing the Gary Seven scenes.
But it can be tricky. The human characters in TOS are basically speaking colloquial English, circa 1966 or so. I have to be careful that my dialogue doesn't feel too dated--or, on the flip side, too jarringly contemporary. And is "on the flip side" a dated reference these days? Could you use it in a TOS novel? I wonder . . . .
Case in point: there's an ep where Kirk jokes about little boys dipping little girls' pigtail in inkwells. That bit was already anachronistic back in the sixties, but would be ridiculously out-of-date today. I doubt you could get away with that in a modern TOS novel.
This is not just a Trek thing, btw. I run into the same issues when writing vintage heroes like The Green Hornet or The Phantom. Recently, when writing the old pulp hero, The Avenger, I went a little overboard on the "Geez, get a load of that big palooka!" dialogue to the extent that my editor had to rein me in a bit . . .