My awkward theory is that when the Soviet Union crumbled, the Reaganaut Republicans took it as their tribal color, in the manner of a globe-spanning psychologically overdetermined game of Capture the Flag.
I can give something like a partial answer here, though I will be interested to see if anyone here can say something at a deeper level.
The gist of it is just that it became a standard during televised election coverage after the advent of widespread access to color television, which then filtered down into everyday usage. But, interestingly enough, the standard took some time to take on its current form. Consider, for example, this electoral map from ABC during the election of '76. Its actually a bit hard for us to read because the red states aren't republican, they are undecided. The yellow states are those that were going for Ford.
This NBC map from the same election is even worse. The colors are reversed! Blue states are republican and red states are democratic! Oddly enough, this means that the solid Democratic south looks, to modern eyes, as republican as it is today. But this is simply an artifact of a time before the association between Republicans and the color red was firmly established. NBC would actually keep using the colors "backwards" at least until 1984.
By '88 though, the networks had settled on the association we know today. Why did they do this? Well, there are some obvious benefits to having some universal standard. For one thing, as it makes electoral maps readable at a glance. Think about how easy it is for you to just glance at an electoral map and know right away which party won which state. When was the last time that you had to look at the key to understand what the colors meant? Standardization made that possible.
As to why these specific colors, I'm speculating a bit here, but it seems likely that they were used because (1) red and blue are primary colors that look good together, (2) that stand out from each other increasing readability, and (3) it is election night, so picking patriotic colors just seems appropriate.
Anyways, as I said by the '88 election the networks settled on the familiar red for republicans and blue for democrats standard. It took a bit longer, of course, for the association to really start to stick in the public consciousness, but eventually seeing these standard maps built up the assumption that red states are republican and blue states are democratic. To my knowledge, it had nothing whatever to do with communism.
