It is certainly true that there was a strong current of socialism in the country in the early 1900s. Mainstream textbooks agree that there were 33 towns and cities with Socialist mayors in 1912. The Socialist Party won roughly 3% of the popular vote in presidential elections in 1904, 1908, 1916 and 1920. It won 6% of the popular vote in 1912. Clearly, this was a movement that was strong in some locales and had a fairly consistent following for about 20 years.
But was this a true driving force in American history as Zinn claims? Zinn sees the Progressive Era as an attempt to appease socialists and he believes that the attack on Mexico in 1914 and the US participation in WWI were simply ways to try to distract people from the fact that they were being oppressed. I'm not convinced. I would argue that the Progressive Era came about because of middle class reformers' desires, not a fear of socialism. I would argue that the US shelled Veracruz out of a combination of arrogance ad impatience with the revolutionary upheaval in Mexico. I do not believe that fear of socialism was a major driving force in the more prominent events of the time. Therefore, I do believe that Zinn is overstating the importance of the socialists.