The usage of the color red for the Republican Party in the US is surprisingly recent. The current color scheme of red and blue didn't become common before the Presidential Elections of 2000. There is a good Washington Post article Red vs. Blue: A history of how we use political colors.
For example, you can see that yellow was used by TV channel ABC for the Republican Party in 1976 (Ford against Carter).
Edit: About the political symbolism of the color red
Today the color red is associated with Communism. This was even more so during the Cold War.
The Soviet Union, (People's Republic of) China, Vietnam and other countries had or still have national flags, which are dominated by red. During the Cold War red was the color of the enemy. Western maps depicted NATO countries in blue and members of the Warsaw Pact in red. Nobody would want red as their color, if they didn't have a good reason (i.e., Socialist and Social Democrat parties - which had already used this color before the Bolshevist Revolution in 1917).
After the disintegration of the Soviet Union and the end of the Cold War it became okay to be associated with the color red. Blue was claimed by the Democrats. Plenty other colors (green, brown, white, purple, ...) have also undesired connotations. Yellow is visually not so attrative - especially in front of a white paper background. So, why not red?
By the way, red is not only the color of Communists. It had been the color of the "flag of defiance". Cities under siege displayed a red flag to show their determination to resist. During the French Revolution the first banner of the revolutionaries was not blue-white-red - but red. The modern French flag became popular as a symbol of reconciliation after French king Louis XVI acceded to the revolutionaries' demands. It merged the red and blue of the city of Paris with the white of the monarch.