The British Invasion knocked most of the folk acts off the charts, but if it hadn’t, they would have fallen off of their own accord. It was a very limited type of music. The Weavers had done practically everything that the folk movement would be known for, and the Kingston Trio hammered it home. Slightly later, Peter, Paul and Mary made their splash largely (but certainly not entirely) doing covers of Dylan songs, and Dylan in his folk phase was mostly recycling Woody Guthrie (and benefitting from the fact that most people had never heard Guthrie).
Mind you, these are all great acts, and there were a number of others (some of which have been mentioned above). But overall there weren’t enough original ideas floating around to sustain a large movement, especially in light of what happened in popular music after the Beatles. I think this is because a lot of the music was so derivative of the work of artists who, themselves, were seldom heard: Leadbelly, Blind Gary Davis, Guthrie, Odetta, etc. It was, in all too many cases, exactly analogous to hearing early rock songs filtered through the ever-so-white voice of Pat Boone. Even the Beatles couldn’t have survived that.
The most enduringly successful musicians (Dylan, Baez, Collins), for the most part, saw the dead end and shuffled off to join the pop mainstream, as did less successful folkies like Roger McGuinn. Some acts, like Peter, Paul and Mary, continued to do well. PP&M were still charting until 1970, when Peter Yarrow decided to make lewd advances on a couple of minors. And they live on every time PBS wants more money.
The story doesn’t quite end there, though, because in 1967 some Brits (especially Fairport Convention and Pentangle, and later, Steeleye Span) managed to connect with their own folk tradition, and so managed to invent British folk rock. British folk rockers have always tended to cross-pollinate with more purist folk acts (e.g., Martin Carthy) and there’s often been a strong jazz influence (going back to Davey Graham, and carrried on by people like Bert Jansch). And a number of straight rock acts also were heavily into folk, most notably Led Zeppelin and Jethro Tull. So this has remained fertile ground in Britain, and even though it’s been in decline since the 80s, it’s never gone away. Thank heaven.