Communism in Hollywood had been something that studio heads and the FBI worried about in the 1930s that intensified in the 1940s. The House of Un-American Activities was not the only force in bringing in the Hollywood Blacklist - the Motion Picture Alliance for the Preservation of American Ideals (which included people from Walt Disney and John Wayne but also many opposed to the Screen Writers Guild) was a group focused on left-wing propaganda they saw in the films made during the Depression and the war years. A culture of loyalty was seen as crucial in the fight against Communism, from the Truman administration down, and many studio heads (like Jack Warner who worked with the FBI clandestinely) were upset over union troubles (the unions went on strike in the mid-40s) that they saw as brought about by Communists.

At this time in Hollywood there was fear of a blacklist, and initially when called to testify before HUAC in 1947, the Committee of the First Amendment was formed of liberal-stars to visit D.C. in support of 19 screenwriters and Communists called up to Washington for hearings. The idea was to stand on the principle of the First Amndment, and that failed with ten witnesses who refused to talk. That's where you get the Hollywood Ten (including Dalton Trumbo) who refused to answer any questions about whether or not they were members of the Communist Party, taking after a well-known Communist Party leader, John Howard Lawson. With contempt of Congress and the publicization of their membership, they quickly lost their liberal allies in Hollywood, either who were upset over their refusal to work with Congress or were forced to denounce them by the studios. The head of the Motion Picture Producers Association declared that the Hollywood Ten would be without a job if they refuse to swear under oath they were not members of the CPUSA, and the blacklist would come about in the years to come.

The Red Scare was quickly dominating American culture by this point. Liberals turned against radicals pretty quickly (the fallout of the Hollywood Ten, the conviction of Alger Hiss and the Rosenberg case soured the CPUSA's reputation) and the Supreme Court legitimized the persecution of the Communist Party. Now that Communists were seen as a threat, the Motion Picture Alliance put the issue on the forefront of what Hollywood should do - House committee meetings brought in those suspected of being members, requiring them to recant and name names in 1951 in order to remain in the film industry. Those who took the Fifth were seen as suspect, and the blacklist took form (200 names) alongside a "graylist" (100 names) created by the American Legion based on groups publicizing stars' and other filmmaker's party ties. Boycotts organized against films had an effect, and directors had to leave Hollywood to find work elsewhere, including those who were never Communists. Movies like The Death of a Salesman, Viva Zapata! and The Day the Earth Stood Still were all targets of criticism. To help stop the boycotts in a struggling industry, Hollywood asked those under criticism (like foreign directors who did not like the way the Hollywood Ten were treated like Billy Wilder and William Wyler) to clear themselves, separating the "liberal lambs from the Communist wolves" in a way suitable to the American Legion.

So I wrote all that to demonstrate the blacklist had a wider affect on Hollywood in general (in a time when Hollywood studios were laying people off already).

Now, as to your feeling about their influence, while screenwriters were not influential in the studio system, there was a indeed a social message found in many films of the Depression, the war and post-war periods. Films like Crossfire (1947) and others dealing with anti-Semitism, and film noir's and other crime films' subtle criticism of the current political order show that whatever remnants of the "Popular Front" between liberals and radicals in the 1930s was still in place by the late 40s. With the decline of domestic audiences, social films did not perform well overseas where studios needed growth to maintain profits, so Hollywood studios wanted to move away from the subject matter and focus more on entertainment.

This shift to entertainment did happen - social problem films didn't win the Best Picture in the early 1950s, and writers turned from social issues to psychological issues in their films. There were more happy endings in the genre of film noirs, less films depicting the upper class or business world as villains, or the focus on youth over adult stories. Writers and directors had to change tack meet both the Cold War pressure and Hollywood's pressure to make content for global audiences or leave - Jules Dassin, Joseph Losey and Cy Endfield all left for Europe for instance. Others also changed the kind of films they made, such as Nicholas Ray, who used the films he made in the 1950s (like Johnny Guitar) as a critique of the blacklist. Some tied social themes and psychological themes, such as Elia Kazan's (a director who had his reputation tarnished because he chose to name names) On the Waterfront and A Face in the Crowd, taking in his 30s idealism with cold realities. The former film was a return of the 1930s common man turned informer against corrupt organized labor and the latter's plot followed a talented common man down on his luck become popular only to turn out to be a egomaniac monster who pushes right-wing politics.

Not everything ended because of the blacklist - films that dealt with racial issues and were considered integrationinst at the time ended because people just stopped seeing them. While studios were wary of the CIA looking for social content in their scripts and films, there also was less opportunities to make lower-profile B-pictures, which were on their way out, and studios focused on drawing back the domestic audience taken in by television. Eventually the blacklist crumbled - filmmakers did manage to make a film outside Hollywood defying the blacklist openly (The Salt of the Earth in 1954) which went under enormous pressure and harassment from the government (they even deported the main star!) Dalton Trumbo's Oscar-award winning screenplay for 1957's The Brave One and changing attitudes helped do away with the system, but not before the blacklist and the intense pressure in Hollywood had changed the system, throwing off many liberal and radical individuals in the film industry that joined in the 1930s off track from a goal of having more artistic control over the content. The ideals of the 1930s, with film culture portraying regular Americans standing up for their own rights against the system and dealing with issues like class conflict were pushed aside in the face that Americans must be united together against a common enemy trying to divide them.

tl;dr: The Hollywood blacklist had about 200 individuals, and the graylist 100 more. It affected far more than those 300 as studios did what they could to not antagonize the government and other watchdogs. Studios themselves participated in the process as did much of the Hollywood establishent as prominent anti-Communists called for a cleansing of the system and liberals began to view Communists as potential foreign agents. With that, social films, of which many films in the 1930s and 40s dealt with issues that faced regular Americans against entrenched systems or elites, pretty much were marginalized by fear or loss of talent by a push for pure entertainment (largely apolitical, but many went about showcasing good American values). Other factors like the loss of domestic audiences to TV also factored in, but the blacklist pretty much put an end for those who wanted to use film to highlight social problems. Hope this answer helps out.