Gradually, between 1920 and the early 1960s
Your assessment is correct — mutants have been around for a long, long time, but there was a sequence of events in the 20th Century that finally brought them to the public's attention. Within the 20th Century, the awareness of mutants seems to have ramped up gradually, between 1920 and the early 1960s.
The following excerpt is from the wiki section of Marvel's official web site. (The bold emphasis is mine.)
While records of Mutants on Earth appear as early as 5,000 years ago (with the emergence of Apocalypse in Egypt) the Mutant population did not grow to significant numbers until the early 20th century. Mutant births, however, began to increase throughout the 20th century (presumably due to environmental factors related to industrialization). In 1920, one of the most prominent mutants was born: Namor the Sub-Mariner, who is considered by many to be the first modern-day Mutant. The Second World War saw the emergence of Mutants as soldiers as several...including Toro, the Sub-Mariner, and Wolverine fought in Allied or Axis armies. The World War II era also saw the beginnings of programs which used mutants as genetic soldiers, such as the Weapon Plus program. Mutant births steeped upwards after the end of the war (following the atomic blasts in Hiroshima and Nagasaki).
As Mutant populations grew, human governments and populations reacted harshly to them, and the Mutants began to form widely divergent ideas about how best to live within the larger human society. Mutants broke off into two rough camps...
(Source: Marvel.com — "Mutants" Article)
Note that Namor's activities were quite "public" — in at least one origin story, he wrecked ships and warred against surface-dwellers and generally attacked human civilization before joining the Allies' fight against the Axis in World War II.
It can't be confirmed, however, to what extent his activities would have been reported to the public-at-large. The answer is likely "not at all", given what we find over at the X-Men Wiki:
Very few people know about mutants before the latter half of the 20th Century...many, if not the vast majority of mutants apparently hid their powers. The general public did know about [supersoldiers] like Captain America. The American military knew of Wolverine, but they were unaware that he was something categorically different from other supersoldiers...Activities of some superheroes and supervillians may have been what notified the general public about the existence of mutants.
(Source: X-Men Wiki — "Mutant" Article)