20 year rule note: This question is specifically about the novel which was published in 1985, rather than the much more recent TV series based on it.
The Handmaid's Tale TV series was considered extremely timely when it began running in 2017, because of the parallels between the ultra-patriarchal regime it depicts and the contemporary online alt-right movement, particularly the parts of that movement that promote ideas like "white sharia". For example, "Our big mistake was teaching [women] to read. We won't make that mistake again." is a quote from the novel that the series was based on, that could easily come from the mouth of any one of a number of alt-right figures who were at the height of their influence in 2017.
However, the novel that the series was based on was first published in 1985, before the Internet as we know it existed, let alone the online alt-right. At the time the novel was published, was anyone actually pushing ideas like prohibiting women from reading? And if so, how fringe or mainstream were such views considered?
(EDIT: removed the geographical qualifier from my question. If women's rights were backsliding to the point of losing the right to read anywhere in the world in the 1980s, I guess that's relevant to the context of the novel's publication)
It wasn't speculative at all.
In a 2019 interview, Atwood was asked about the contents of her archives. The interview was organizing around the bigger idea of what inspired Atwood's writing. From the interview:
Q: What are all these newspaper clippings?
MA: They are Handmaid’s Tale background material. They’re nicely sorted and laminated. ‘Women forced to have babies.’ This is an article about Ceaușescu and Romania. He passed laws that said women had to have four babies. They had to have pregnancy tests every month and if they weren’t pregnant they had to explain why. ‘The latest sicko Red ruling was announced by cold-blooded Romanian president Nicolas [sic] Ceaușescu, who wants women to have more babies so the country will get richer.’ It was this policy that filled up the Romanian orphanages, which then became a scandal around the world for their inhumane conditions.
Here’s another piece: ‘Conservatives are out to get the women’s movement. They wish to attack birth control and voluntary sterilization. Their eventual target is to wipe out the women’s movement.’ And this is a good headline that highlights religious tensions: ‘Catholics say cult taking over.’ It’s about a cult called the People of Hope that ‘subordinates its women, discourages social contact with non-members, arranges marriages, moves teenage disciples to households for indoctrination . . . their treatment of women is very Islamic. It’s a form of brainwashing.’
Later in the interview:
Q: It must have been very intense researching this way.
MA: I didn’t even research it. There was no Internet then, you couldn’t just go online and put in a topic, so this is just stuff I came across when reading newspapers and magazines. I cut things out and put them in a box. I already knew what I was writing about and this was backup. In case someone said, ‘How did you make this up?’ As I’ve said about a million times, I didn’t make it up. This is the proof – everything in these boxes.
There’s so much of it. ‘Ayatollah Khomeini’s slaughterhouse. The jails of Iran.’ Here’s one about Klan-watch, an organization that tracked the Ku Klux Klan. White supremacists were very active then. ‘Liberation theology, why is the Vatican so worried?’ Death row. The battle of the Bible belt. In 1985 racists in the US were preaching revolution: ‘US conservatives push new order – the push to the right.’ And religion was a significant part of that whole movement: ‘Racism and religion a potent brew,’ ‘Evangelist moles running for US president,’ ‘The power and influence behind America’s right’ – meaning the rise of the religious right.
Talking of the rise of the right, here’s a piece about the Lebensborn movement in Nazi Germany, when SS men were given racially pure extra ‘wives’ to make more little SS men: ‘Why Nazis slaughtered own super-race babies . . . The super-race child would be bred for looks and loyalty, stalwart, tall, blond-haired and blue-eyed. Nazism would be his creed, Adolf Hitler his god...’
Basically, she took a number of stories from around the world and combined them into Gilead. I'd recommend the whole article as it goes into more details regarding her inspiration. Meanwhile, in other interviews, Atwood made connections to the lived experiences of enslaved women in the Antebellum United States which can shed some light on the matter of women's rights backsliding to the point of losing the right to read. Regarding that particular point, I would recommend a different article, "Both versions of The Handmaid’s Tale have a problem with racial erasure":
In Western fiction, dystopic stories often ask, "What if this atrocity had happened to white people instead?"
That was the formula more than 100 years ago, when H.G. Wells wrote The War of the Worlds, its narrator comparing the Martian invasion of Britain to Britain’s ruthless invasion of Tasmania. It’s the formula in Universal Pictures’ new franchise-starter The Mummy, which envisions a Middle Eastern woman bringing war to London, as Brits and Americans brought war to Iraq. And it’s the basis for Margaret Atwood's 1985 novel The Handmaid's Tale, which imagines a world in which white women are enslaved and sexually coerced as black women were under American slavery.
An example of this is related to your question: in the Antebellum United States, a free Black woman who traveled from the North to the South would lose her right to, or teach her children to, read. While the exact specifics would vary depending on state (some states didn't outlaw literacy as much as they outlawed formal instruction for enslaved children and adults), it was functionally the same as what Atwood described in her book.