UK Cabinet papers from 1986, released by the National Archives this week, reveal Thatcher repeatedly queried the wording of advertisements and leaflets.
Writing on a memo describing the campaign, Thatcher wrote: "Do we have to have the section on risky sex? I should have thought it could do immense harm if young teenagers were to read it?"
The arrival of the disease in Britain in the early 1980s had persuaded ministers that urgent action was needed to prevent the spread of HIV with an unprecedented public education campaign outlining its dangers and how to prevent infection.
Another document released by archive shows Thatcher's reservations were put forward by her deputy, Lord Whitelaw, who was chairing a Cabinet sub-committee discussing Fowler's text the following day.
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"As there was no support at all for the doubts I had aired, the committee agreed that the publicity should go ahead next month," he later told her.
"I remain against certain parts of this advertisement. I think the anxiety on the part of parents and many teenagers who would never be in danger from AIDS exceeds the good it may do... Adverts where every young person will read and hear of practices they never knew about will do harm," she responded.
However, there was massive ignorance about how it spread and there were an unknown number of people who had the disease but did not know.
The row ended when health officials suggested that the phrase "anal sex" be replaced with "rectal intercourse" and Thatcher finally declared the new language acceptable.
Although criticised at the time for scaremongering, the campaign was later recognised to have been one of the most successful in the world and limited HIV infections in Britain to around half the level of other European countries.
Thatcher was Prime Minister of the UK from 1979 to 1990, the first woman to hold the position. She died in 2013.