This is the second in a series of Mischiefs of Faction posts about the politics of Star Wars, leading up to the release of Star Wars: The Force Awakens. In the first post in our series, Seth Masket argued that the Galactic Republic fell because it lacked a strong Senate minority party, which could have more closely scrutinized Chancellor Palpatine as he consolidated power and turned himself into an emperor.
While I agree with Seth's point, I think an even bigger problem was the role of the Jedi. It is a bad idea for a republic to outsource its police and military power, as well as most of its diplomacy, to an autonomous religious cult. Monopoly on the use of force is a central function of a healthy state. The Galactic Republic relied on the Jedi to enforce its will domestically and internationally. Such a scenario made the republic very vulnerable to a Jedi coup, something senators would have been aware of and vigilant against. The Senate was insufficiently vigilant against a rising Emperor Palpatine because its main fear was an Emperor Yoda.
All governments need a monopoly on the use of force. A sign of an unstable republic is when the military and police are not subordinate to civilian political institutions. Around the world, countries where military members are more loyal to each other and to their generals than to civilian politicians tend to be dictatorships or fragile democracies that are vulnerable to coups.
The fall of the Galactic Republic is clearly inspired by the history of ancient Rome. The Roman Republic became increasingly unstable when the Roman legionnaires became loyal to one another and their generals rather than the Senate. It wasn't long before successful generals took the opportunity to take over the government and make themselves emperor, a pattern that ended the Roman Republic and repeated itself over and over during Roman Empire.
Similarly, a key turning point during the French First Republic was, during the war of the first coalition, allowing the French armies to "live off the land." Rather than relying on the government in Paris for their pay, French armies were paid from resources looted or taxed from territories they conquered. Soldiers' loyalties quickly shifted to the generals who produced their paychecks, especially to General Napoleon, who conquered the wealthy regions of Italy and Switzerland, compensating his troops generously. In the US, a key part of our government's overall stability rests on civilian control of the military, which comes primarily from a strong culture of subordination to civilian leadership within the armed forces and complete control of the military budget by Congress.
Like many aspects of the Galactic Republic's political structure, the exact relationship between the Jedi and the Senate is left vague. Leading Jedi do consult in the movies with the chancellor (who is elected by the Senate). But the Jedi have a parallel leadership structure, the Jedi Council, and are socialized to be at least as loyal to it as to the Senate.
Given this situation, it was actually a good idea for the Senate and chancellor to raise a clone army that would be loyal primarily to the political branches of the government. Relying on the Jedi was a recipe for constant instability and fear of a coup. If the Republic had successfully transitioned to relying primarily on the clone (i.e., Stormtrooper) army and the Jedi became less involved in politics, this would have produced a more stable system.
But before that transition happened, Chancellor Palpatine exploited the still real danger of Jedi interference in politics to take dictatorial control for himself, as he had planned. When Palpatine went to the Senate and said the Jedi had attempted a coup, this was actually plausible. The senators still needed to be constantly vigilant against this scenario. If they hadn't been in a system where they had to fear a Jedi coup, they might have been more cautious about ceding emergency dictatorial power to the chancellor.
Extra note: The precise constitutional structure of the Galactic Republic is left vague in the movies in several important ways. (And no, I refuse to read anything about the expanded universe.) But according to Wookieepedia, the chancellor is elected to the Senate for terms that last a maximum of four years and is term-limited to no more than two of those terms. The chancellor can lose power before his or her term is up though a Senate vote of no confidence, but it is unclear whether the chancellor can call early Senate elections, as prime ministers can in most parliamentary democracies. Even with that uncertainty, the chancellorship seems clearly designed to be an expression of the Senate's will. So for all its problems, the Galactic Republic wasn't vulnerable to the typical problems of presidential systems.