Disclaimer: I’m a Mason and I know the general things we say, but I’d be interested in a hopefully unbiased look at the question from a historical perspective.
EDIT: Words are hard/spelling
Hey, I think you would be interested in my /r/AskSocialScience answer:
The whole answer is worth reading (indeed, the whole article it's based on, Robert Putnam's "Bowling Alone", is worth reading, and maybe even the longer book version). Putnam argues that it's civic organizations of all kinds that have seen declined participation over the past few decades, from bowling leagues to churches to elks lodges.
He offers two answers: demographic changes that reduce civic engagement (mostly things that weaken the family) and the "technological transformation of leisure" (writing in the 90's, he's mostly talking about the television, but the internet has exacerbated this considerably).
He also rejects two hypotheses that others have offered: women entering the workforce (movement of women into the workforce) and increased geographical mobility (the "re-potting" hypothesis, where people who move frequently form weaker roots). I discuss all four of these in more detail in the post linked above.
Personally, I think his look at changing family structure is too narrow, and I suspect that there is an increase in the amount of time spouses now expect to spend with each other. I think there has been some degree a refocusing on the family--parents might go to bowling leagues less, but they probably go to kids' soccer leagues more. Additionally, I don't think he doesn't spend enough time thinking about changes in the kind of work men and women do (especially the middle class and upper class men and women who tend to join Elks lodges).
Putnam also has a later line of research that showed increased diversity decreases social trust (probably just over the short term, not the long term, he argues I think convincingly) and that might also have an effect on institutional participation. See here for a discussion of that. Findings about homophily ("birds of a feather flock together") are one of the most consistent findings from modern sociology, so this is perhaps less surprising than you'd think. See here. I don't think it actually explains that much of the decline in social organizations, as they declined even in areas that saw relatively little changes in racial, ethnic, religious make up, as well. This just frequently comes up when this question is asked, particularly because Putnam wrote both.
Putnam's article was a "break out hit", at least by social science standards. It was highly debated. One argument that came out of this is that people have changed how they socialize, but it's not clear they socialize considerably less. Elks lodges have declined, sure, but the Elks, for example, only date to 1868 (I have a soft spot for the Elks because I went to a million punk rock shows at the Elks Lodge in Cambridge, MA). Two hundred years ago, fraternal societies were not as important--things like guilds still were very important in cities, things like maypole dances were important in the countryside. Elks lodges' heyday may have passed, but not we have different forms of socialization like, well, Reddit and other internet-y things. I think this is true, and this topic is much debated, but it does seem like overall like the proportion of socially isolated people has have decreased over the past few decades as well.
Some criticisms of Putnam, I should add, include that sociologists have been worried about some sort of decline of socialization since as long as sociology has existed. Karl Marx worried about alienation. Émile Durkheim was responding to the decline of traditional village communities (what he called "mechanical solidarity") where everyone was tied to everyone and the rise of more anonymous urban living (what he called "organic solidarity"). Several early ethnographies, like the Middletown study of the 1920's, also argued that people felt a decline in socialization and "neighborly feelings", this time specifically pointing to isolating entertainment technology, just radio instead of TV or the internet. This doesn't mean that these things aren't true, I just wanted to point that this has been a perennial concern of sociology.
In short, the decline of Freemasons and the Elks was part of a larger secular decline in community organizations. The exact reasons for this overall decline are highly debated, but seem to be related to other changes in American life, including changes in preferences for how to socialize, changes in technology especial in terms of leisure, and changes in demographics especially in terms of changing family structures. There is, unfortunately, no one clear answer. For people who want a good critique of Putnam's work, I recommend Berkeley demographer Claude Fischer's longer review from 2005 called "Bowling Alone: What's the Score"? You can find a PDF here. It was originally presented at the "Author Meets Critic" session of the American Sociological Association. Despite Fischer's valid criticism, I do think it is safe to say the first two points of this paragraph: there's been a clear decline in fraternal organizations, and it's part of a wider trend in decreasing formal membership in similar organizations since about 1970. This was first noted in the 1990's, and the trend has continued apace.
Can you elaborate on 'the general things we say'? I for one would love to know what the internal position is and it might help spur some responses.
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Edit: To the Anon reporter -
1: Ping /u/yodatsracist on Bowling Alone