The era before computers is a long one...! So I'll offer thoughts only on two different eras.
First, the 1890s:
In my own historical research, I've come across classified ads in cheap newspapers of the 1890s offering "Dissertations: Correctly Annotated and Formatted" for sale. (!!!) Really threw me when I saw that one, but I guess the desire to plagiarize is timeless.
However, this type of thing would have likely been very difficult to pass off on a dissertation committee. First, they are experts in the field, and have (theoretically) been conversing with their (would-be plagiarizing) doctoral student on a regular basis. They are almost certain to spot some radically-new direction that their student has 'taken' when the dissertation just magically appears, complete and correctly annotated.
The scope of a field was much smaller as well. For instance, a historian of modern Britain in the 1890s could very well have read every single book and article in their field over the course of their career. (which would make it easy to spot plagiarism... "hey, I've seen that argument before!") These days, however--with the explosion of research publishing over the last 100 years--it is simply impossible for a historian to read everything in our own field... and even difficult to keep current in one's own subfield!
And remember three things about undergrad teaching in the 1890s:
class sizes tended to be a lot smaller. So teachers knew their students--and their students' capabilities--a lot better. Which made cheating tougher.
those who attended university in the 1890s were almost certainly of the elite... which means that they were better prepared (educated) in the first place, and had a lot to lose if their honor were challenged and/or they were caught cheating
those who would plagiarize would (by definition) be late and/or lazy... and there were few resources available to the late-and-lazy in the 1890s. How do you find the text to plagiarize?
You could copy a section out of a book, but the professor was likely quite familiar with the books in your library in his area of expertise... pretty easy to catch.
Instead, the major form of plagiarism in this era would be for a busy wealthy kid to pay someone to write their papers for them... and this certainly happened. But it requires that a willing 'ghost-writer' be found... which is really what the classified ad was offering to do. (i.e. write the dissertation 'on spec')
Finding these willing writers was tough, and put inherent limitations on how far the cheating can go. i.e. If you're a professor teaching a class of 30, maybe one student could find one other to pay to write him a paper... but certainly not all 30.
A quick digression: outside of the ivy-covered halls of the university, on the other hand, plagiarism of news was rampant. In fact, in my era (late 19th century), most newspapers and magazines freely plagiarized from other newspapers and magazines, often publishing it as their own reporting. This was especially true trans-nationally, i.e. an American paper would basically copy articles from a London magazine, and Germans would just translate from an American magazine, and all would pass it off as their own research. If you read widely on a topic across many newspapers, large and small, in the 1890s, you'll see an absolute TON of copying, often verbatim, and mostly unattributed. This, to continue my digression, is my major beef with relying on Google N-Grams; it will be totally fooled, mistaking mere copying as popularity of usage. But I digress.
1980s (i.e. pre-internet)
While the social 'reach' of the university expanded dramatically in the late '40s (GI Bill) and even more dramatically in the '60s, so you have a LOT more 'range' in students' preparedness. Namely, you now have a very large number of students who, because they come from a broader social strata, could never have "made it" to (or in) the rarified, more-demanding university setting of the 1890s.
However, with people from a much greater variety of backgrounds coming into college, professors faced a less coherent student body, with less consistent educational foundation.
Two things happened: standards (and workloads) dropped, and other majors appeared. (The other majors were more professional training, like physiology or communications or hotel management. The number of History majors, for instance, has actually been fairly consistent across the last 60 years in absolute terms... it's only as percentage of total university degrees awarded where we see History majors decline, relatively... but I digress.)
So, what does this mean for plagiarism? Well, with students coming in far less prepared, there was a lot more incentive (and perceived need) to plagiarize.
And with classes (much) larger (in the 200s or more), and book- and article-publication having exploded past the point where a professor might have read everything, there was more opportunity to get away with it.
The real bottleneck remained one of supply. If 30 or even 50 students in a 200 person class wanted (or felt that they needed) to plagiarize, they would still have to find text the "old fashioned way", namely, that meant finding a book or an article to copy out of. So, 30 would-be-plagiarizers would need to find 30 different books and/or articles to plagiarize out of. Often, the search for suitable plagiarism could lead to actual research...! :-) But when the average writing and research skills of their students weaker (relatively) than in the 1890s, spotting copying from books was fairly easy.
Indeed, by the 1980s, the 'gap' between what the students could find in the library and what they could hope to pass off as their own work had widened considerably: a mediocre studentwho copied from a book, article or textbook would be easily caught, as it was (now) unlikely that they could actually write that well.
You could (and did) have papers "recycled" from seniors to freshman, of course. There's a great scene in the film Animal House (if I remember correctly) which shows the fraternity's filing cabinet of past papers and past exams. Of course, Ye Old Fraternity Filing Cabinet only "works" if a professor doesn't change his/her exams or paper topics often. So again, there was a bit of a constraint on cheating and plagiarism there, in that, if a moderately diligent professor changed an assignment somewhat, it would instantly obsolete the filing-cabinet shortcut.
You still had students pay other students to write papers for them--and indeed, in some fraternities, "smart" students were let in specifically to be 'resources' for their brethren.
But still, a class of 200 is not going to find 50 students willing to write their papers for them. And going to the library to dig up your own sources to copy was almost as taxing as doing the research in the first place.
Enter the Internet:
What has turned plagiarism from a minor problem of a few bad apples (in previous centuries) into a scourge that is threatening to destroy the very process of higher education is simply the internet: namely, crowd-sourcing cheating, and (especially) cheating encouraged by for-profit entities.
Now, students upload their old papers... but they do so by the millions. So, there is no way to 'change' an assignment... because however you design it, there is likely a half-dozen papers floating out there fairly close to that topic. (I know--I've tried. and tried.)
And there's just so much more text available with the internet. Crappy text, true, but text nonetheless.
Finally, there are literally hundreds of companies offering to sell you a paper--on any topic--right there, pushing their "wares" on you right when you're most desperate--scrounging the internet for a germ of an idea. (and advertising works...!)
This, I would argue, fundamentally threatens the entire edifice of a university education. If I think there's a good chance a paper is plagiarized, there's no incentive for me to bother commenting on or correcting it. If a student plagiarizes a lot of papers, they never learn to write... but still get the degree. So then their would-be employer realizes that a university degree does NOT mean that their hiree is literate. And so will place less value on that degree. And so begins the spiral down....
Thankfully, technology caught up: plagiarism detectors (TurnItIn) have gotten very good. Yes, there are Synonymizers--sites where, after the plagiarizer has downloaded a paper which would be caught by the detector, they then feed the plagiarized text into a program that replaces a certain percentage of words with their synonym, which does indeed the plagiarism undetectable. Unfortunately for the plagiarizer, it also makes the paper virtually unreadable. (I get about 2 of these per semester in a 200 person class... I can't "prove" plagiarism, of course, but I can prove horrible writing, so they get Ds.)
Still, I long for the days of teaching pre-internet. Back when you didn't have this quasi-combatitive relationship with your large classes, where you knew that about 30% of them were going to try to cheat... which just sucks the fun out of teaching.
EDIT: once computer-generated writing, and computer-led research, gets sophisticated enough to research and write a paper that I cannot tell has been written and researched by a machine, then our educational system will be in a major crisis. I see everything going back to oral- and in-person timed exams.