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There have been a number of good assessments of how the way in which we depict the two parties has evolved. One of the earliest appeared here in the Post, shortly before the 2004 election. There was one shortly after that in Washington Monthly which is often cited; probably the most thorough is one, from the Smithsonian.
What's clear is that, prior to 2000, talking about "red states" and "blue states" wouldn't necessarily have resulted in any understanding. In 1992, David Nyhan of the Boston Globe wrote of his mixed feelings about Bill Clinton's candidacy. "[W]hen the anchormen turn to their electronic tote boards election night," he wrote, "and the red states for Clinton start swamping the blue states for Bush, this will be a strange night for me." You'll notice that those colors are backward, by our current understanding. Nyhan is being figurative we can assume, recognizing the standard red-blue split if not the significance of the colors.
But by 1992, networks seem to have mostly settled on red-for-Republican, blue-for-Democrat. 
The Smithsonian article tells a great story.  
There's some discussion of why networks first drifted toward those colors. Red, the Smithsonian article suggests, was color non grata in American politics at the time, thanks to the Cold War. Red, of course, was associated with the Soviet Union, because it was associated with communism and socialism -- and with left-leaning parties internationally. (See this poll from the  British elections: Labour is red; the Conservatives, blue.)
Most Americans probably weren't paying very close attention to the decisions made by networks every four years, so it's pretty safe to assume that they wouldn't immediately have associated the parties as we do now.
Then there was the election of 2000, which, if you were born at any point after, say, 1985, and care even a tiny bit about politics, is seared into your memory. This was just at the cusp of the internet/cable news era, so pundits kept busy making and pointing to and discussing maps of the states and how Bush or Gore would end up winning this thing. And those maps, in heavy rotation, looked like the maps from election night: Republican red, Democratic blue.  
The 2004 Post article by Paul Farhi suggests that "the 2000 election, NBC's graphics department and David Letterman all played critical roles." NBC graphics, because that prompted Tim Russert and Matt Lauer to discuss red and blue states shortly before Election Night. David Letterman, because he made one of the early jokes on the subject, while Florida's votes were still being counted. "The candidates will work out a compromise," he joked on November 14. "And thank God, not a minute too soon. Here's how it's going to go. George W. Bush will be president for the red states. Al W. Gore will be president for the blue states. And that's -- that's the best they can do."
Two things are interesting about that. First, that the audience is expected to understand what it means. And second, that Letterman was hardly the first to make the joke. A letter to the editors of the Post on November 11 from Robert Forman made the same proposal, "Why not let Gore be the president of the United States of America to include all the blue states east of the Mississippi? Bush could be the president of the Confederate States of America, to include all the red states on both sides of the Mississippi." The dichotomy had already set in.  
It's become so familiar that there's no reason to think it will go away in our ever-so-polarized politics.   

It is surprising to me that the liberal rag Washington Post would mention, even in passing that red was the traditional color of leftist liberal/socialist/communist politics.  It is not surprising that such libs as Russert and Lauer would flip flop the colors to disassociate their beloved Demoncat party from its fascist/socialist foundations.
I do appreciate and admire Dave Leip for assigning the traditional/historic colors to the parties in his Atlas of US Presidential Elections.