Quick answer: women began joining cheerleading in the 1920s - the University of Minnesota first allowed women to cheer in 1923 - but the major shift toward women's participation is generally linked to the 1940s, when many male college students were away during the Second World War - after which cheerleading gradually became a predominantly female activity.
Early, Limited Participation
In cheerleading's first few decades, the activity was almost entirely performed by men - see Were the First Cheerleaders Men?. The University of Minnesota, where organised cheering is usually said to have begun, first allowed women to cheer in 1923. Other colleges gradually followed, but through the 1920s and 1930s women remained the exception rather than the norm.
The Wartime Shift
The most commonly cited turning point is the 1940s: with large numbers of male college students away during the Second World War, women stepped into cheerleading roles in much greater numbers, and the activity's gender balance began to shift.
Why the Shift Stuck
Once women were established in cheerleading roles during the 1940s, participation continued to grow after the war rather than reverting - over the following decades, cheerleading became increasingly associated with young women, particularly at the school level, a pattern that has remained broadly true since.
A More Complex Picture Today
While cheerleading has become predominantly associated with women, particularly in school and sideline contexts, men remain part of the sport - especially in All Star and competitive cheer, where mixed-gender ("coed") teams are common and male athletes are often valued specifically for stunting strength. See What Is Coed Cheer? for more.
A Note on Dates
Some early details vary slightly between sources. Where sources disagree, we give the most widely cited version and avoid false precision - the sources we used are listed below.
Sources: Cheerleading (Wikipedia); Being a Cheerleader - History of Cheerleading (Varsity)