Is attacking downhills with coaster brakes where the sport of downhill mountain biking originated? Or did mountain biking and klunking grow up side by side?
Klunker as a term is synonymous with early mountain bike (as it is currently known). Klunkers were developed by customising early beach cruisers on 26" wheels (Schwinn predominantly) by spreading the rear dropouts to make them multi-geared and adding moto handle bars and brakes due to their durability and strength (brands such as Magura were initially moto brands).
The term klunker was used to describe these built up bikes on cruiser frames.
This style of bike and mountain riding first appeared in Marin County, California in the mid 70's prior to the first purpose designed mountain bike appearing in 78. If you look at the proponents of the early mountain biking movement and design; Gary Fisher, Joe Breeze, Charlie Kelly, Tom Ritchey and Keith Bontrager (there are going to be many more....) all were active in Marin and it's surrounding counties at the time.
The story is complicated and can't be confined to a couple of paragraphs. That's why I wrote a book about it: Fat Tire Flyer: Repack and the Birth of Mountain Biking.
Hard to be definitive. But I thought it started with Gary Fisher and others on cruiser bikes with coaster brakes https://en.wikipedia.org/wiki/Gary_Fisher, the Repack Races. He went on to develop the first mountain bikes.
I have since discovered that the answer is indeed yes. Men like Joe Breeze and Gary Fisher tools their old 50s and 60s schwinns and added balloon tires and two brakes, after the Repack left many with smoking coaster brakes. Tom Ritchey made the first mountain bike for Joe Breeze, and they eventually built the first mountain bikes or sale as well as coin the term. In 1981 the first mass production bike came from Specialized with the Stumpjumper. What started as coaster braking a cruiser down some dirt path eventually became an Olympic sport. While, as Daniel pointed out, bikes had been ridden off road way before that, it was these men that invented the word and the sport.