In preparation for the famed March on Washington where MLK makes his "I Have a Dream" speech, a committee was formed months in advance to coordinate bus drop offs at the National Mall. Buses picked people up from around the country and brought them to the Mall at a set time to drop people off and pick them back up. It actually worked fairly well.

Before that, for protests like the Bonus Army march after WWI, they'd get there by train or foot. Since these protests built up over time, traffic in getting there was less of a concern.

So in short, a lot like they do today, but admittedly with fewer transportation options (not like Metro in DC is all that reliable!).