Well, not so much, according to a new NBER working paper by Gentzkow, Shapiro, and Sinkinson (ungated version here):
We use new data on entries and exits of US daily newspapers from 1869 to 2004 to estimate effects on political participation, party vote shares, and electoral competitiveness. Our identification strategy exploits the precise timing of these events and allows for the possibility of confounding trends. We find that newspapers have a robust positive effect on political participation, with one additional newspaper increasing both presidential and congressional turnout by approximately 0.3 percentage points. Newspaper competition is not a key driver of turnout: our effect is driven mainly by the first newspaper in a market, and the effect of a second or third paper is significantly smaller. The effect on presidential turnout diminishes after the introduction of radio and television, while the estimated effect on congressional turnout remains similar up to recent years. We find no evidence that partisan newspapers affect party vote shares, with confidence intervals that rule out even moderate-sized effects. We find no clear evidence that newspapers systematically help or hurt incumbents.
So to sum up, newspapers don't affect the composition of the vote and since the introduction of radio and then television (i.e. for a long, long time) have only a very small influence on the size of the vote.
Hey, but at least they do have Dilbert!