Like the United Nations, the Republic has no armed forces of its own, but instead relies on a group of warriors, the Jedi knights, to "keep the peace." The Jedi, while autonomous, often work in tandem with the Senate, trying to smooth over quarrels and avoid conflicts. But the Jedi number only in the thousands--they cannot protect everyone.
What's more, it's not clear that they should be "protecting" anyone. The Jedi are Lucas's great heroes, full of Zen wisdom and righteous power. They encourage people to "use the Force"--the mystical energy which is the source of their power--but the truth, revealed in "The Phantom Menace," is that the Force isn't available to the rabble. The Force comes from midi-chlorians, tiny symbiotic organisms in people's blood, like mitochondria. The Force, it turns out, is an inherited, genetic trait. If you don't have the blood, you don't get the Force. Which makes the Jedi not a democratic militia, but a royalist Swiss guard....
In all of the time we spend observing the Rebel Alliance, we never hear of their governing strategy or their plans for a post-Imperial universe. All we see are plots and fighting. Their victory over the Empire doesn't liberate the galaxy--it turns the galaxy into Somalia writ large: dominated by local warlords who are answerable to no one