A few days ago, the New York Times had a story called "Boxing is a brutal, fading sport. Could football be next?" It pointed out that boxing used to be very popular, but said "there came a time when the fight game’s hold on the American spirit began to loosen, when it stood widely condemned as plain brutal." As the title suggests, it went on to say that maybe the same thing was starting to happen to football.
I looked for questions about whether boxing should be banned, and found a number going back to the 1950s. Several gave a simple choice:
Banned Not DK
6/1955 39% 45% 16%
4/1962 37% 47% 16%
4/1963 41% 43% 16%
6/1965 43% 45% 12%
3/1981 23% 70% 7%
7/1997 19% 74% 7%
One in 1984 gave people a choice of banning only amateur boxing, only professional boxing, or banning both. 39% favored both, 8% favored banning one of them, and 50% said both should be allowed. In 1999, a Fox survey gave a choice between regulated better (42%), left alone (24%), or abolished altogether (24%).
Public opinion didn't turn against boxing in a straightforward sense--in fact, support for abolishing boxing was generally lower in the 1980s and 1990s. Or looking at it another way, boxing's hold on the American spirit wasn't secure even in its glory days: about 40% wanted it abolished.
There are no questions about whether football should be abolished, but several in the past few years have asked if the respondent would allow a son to play football: about 70-75% say they would. People do seem to be aware of the dangers of football: in 2011, a survey asked "in which one of the following professional sports do you think athletes are most likely to sustain a permanent injury that will affect them after they retire?...NFL (National Football League) football, martial arts, boxing, hockey." More chose football (49%) than boxing (36%).
The lesson is that changes in the fortunes of the sports didn't directly reflect changes in public opinion. I think the basic reason that football flourished while boxing declined is that football developed an organization that could supply a consistent "product," and boxing never did.
[Source: iPOLL, Roper Center for Public Opinion Research. The Roper Center has moved from UConn, where it had been since 1977, to Cornell University. I'm sorry to see it go (I was interim director from 2004-6), but I'm glad that it found what promises to be a good home.]