According to the economists Stefano DellaVigna of U.C.-Berkeley and Ethan Kaplan of Stockholm University, Fox News has a direct influence on election results. The beneficiaries, as you'd expect, are Republicans.
Their research, published by the National Bureau of Economic Research, examined the differences in 1996 and 2000 election results in areas where Fox News first became available between the two elections. From their study of more than 9,000 towns in 28 states they found that "Republicans gain 0.4 to 0.7 percentage points in the towns which broadcast Fox News.…We also find a significant effect of Fox News on Senate vote share and on voter turnout. Our estimates imply that Fox News convinced 3 to 8 percent of its viewers to vote Republican." (Their assumption that Fox has a GOP bias is based on one cited paper and not detailed in their own.)
The authors calculate that Fox pulled 200,000 votes nationally to the GOP and break down the apparent influence more specifically: "Fox News had a smaller effect in rural areas, in Republican congressional districts, and in the South," where more people already voted Republican. Fox's apparent effect was also "smaller in towns with more cable channels, consistent with competition reducing the media effect."
Since Fox appeared to have a strong effect on votes for Senate races, where the station was usually not directly covering the specific candidates, the economists presume the effect was ideological, not candidate-specific. They also found Fox's influence had more to do with inducing nonvoters to hit the booth than with luring existing voters across party lines.
The authors think their research shows a disturbing suggestibility in American voters. But attempts to persuade your fellow citizens are the very marrow of political life. For those who think the station is too influential, the American response isn't less Fox speech. It's more anti-Fox speech.