Below we plot the labor force participation rate of men and women over time. The share of women who are either working or looking for work has nearly doubled since 1948, while the share of men has fallen by nearly 20 percent in the same time-period.
Despite the entrance of women into the labor market, the unemployment rate of men has hardly changed over time.
The huge increase in competition for work was not met with any notable difficulty finding work among men. In general, a larger labor force does not imply more difficulty finding work. This holds both when women enter the labor market and when immigrants enter the labor market.