A martini was indeed a high class drink going back to the 1920s. After various experimentations and variations the martini became more regularised in London cocktail bars, and from there spread around the world as a 2:1 mix of gin and vermouth (botanical fortified wine - note there is now a popular brand of vermouth called "Martini" but "a martini" is not Martini, it's a cocktail - although it can use Martini brand vermouth in it) garnished with an olive. Then throughout WW2 France and Italy being off limits meant that it was very hard to get hold of vermouth. The result of rationing and also the tastes of the day meant that martinis became drier and drier (ie more and more gin and less and less vermouth). Winston Churchill (edit: Noël Coward - see comments) once said the perfect mix for a martini was to fill a glass with gin and then wave it in the direction of France.

By the 1950s with rationing over martinis could become wetter again. At this point the fashion became to experiment and try new things to spice the martini up a bit. So you get vodka martinis (vodka instead of gin), dirty martinis (a martini with olive brine added to it) etc... In James Bond Ian Fleming has Bond drink all sorts of different drinks to show what a sophisticate and alcoholic he is. He drinks mostly champagne and whisky soda, but he also works his way through many cocktail menus and drinks a lot of martinis and vodka martinis, and also - in Casino Royale in 1953 - invents his own: the "Vesper Martini" which is 6:2:1 gin, vodka and Kina Lillet (which is a lemony quininey vermouth) finished with a twist of lemon.

Then for the films starting with Dr No in 1962 the Broccolis arranged a product placement with Smirnoff Vodka (and Red Stripe beer) and so in the films Bond mostly drinks vodka martinis, although in later films they add in product placement for other whiskies, beers and champagnes too.

So to answer your question the martini was high class a bit earlier than the 50s and 60s, in the 50s and 60s what was high class was experimentation, which is what book Bond does. Then film Bond drinks vodka martini as part of a brand deal and in so doing becomes a taste maker as much as a taste follower and establishes the martini as high class.

Source is mostly things I remembered from a fascinating talk on the history of gin at the London Gin Club but I also looked up a few things (this was interesting for example) and I have myself being doing some research on the cultural signifiers in James Bond for a book I wrote a first draft of 2 years ago and still haven't finished.