In a recent article, Thomas Edsall gave a graph showing that the percentage of people saying that religion was losing influence had increased substantially between 2012 and 2014. I thought that the question went farther back than that, so I searched for it. I got more than I bargained for, because it was asked no less than 78 times between 1957 and 2014, first by Gallup, which still asks it from time to time, and then by other organizations, notably Pew. The exact question is: "At the present time, do you think religion as a whole is increasing its influence on American life or losing its influence?" Sometimes people volunteered that it was staying the same, or some other intermediate answer like "some of both," but those number varied widely (from 1% to 17%), so I think that organizations must have differed in the extent to which they pressed for an answer. To take them out of the picture, I took the odds of "increasing" to "losing." For example, the last time it was asked, 22% said increasing and 72% said losing, for odds of about 1:3.3. Here are the results over time (shown on a log scale):
There are a couple of obvious changes: a big shift towards "losing influence" in the 1960s and a large but short-lived shift towards "increasing influence" after 9/11. But there are also several other shifts that took place over a substantial period of time: towards "losing influence" from the mid-1980s to the early/mid 1990s, then back towards "increasing influence until about 2000, then towards "losing influence" since 2002. I can't think of a plausible reason for them: they don't seem to follow the political climate, or trends in things that people might think of as moral problems like the crime rate or divorce. But given the amount of data here, they're not just chance variation.
[Data from iPOLL, Roper Center for Public Opinion Research]