The Galactic Republic was corrupt. Many of its Senators and bureaucrats were corrupt. That became clear when the Senate would not lift a tentacle to save Naboo from attack by the Trade Federation when its people were dying.
Surely the Jedi knew the Republic was a democracy in name only, and that it was hopelessly corrupt. It was probably corrupt centuries before a politician named Palpatine began manipulating the Trade Federation and Naboo towards war.
They surely knew - or should have known - the inner worlds became wealthy by oppressing the outer rim worlds. This oppression led to many outer rim worlds breaking away from the Republic to create the Confederacy of Independent Systems.
By supporting it as peacekeepers and negotiators, the Jedi helped prolong the suffering of people affected by corrupt governments and corporations.
Why would the idealistic Jedi serve a corrupt system?
Edit: I am offering a bounty for an answer because none of the answers so far really satisfy the question. TheIronCheek provided a really good reply that my premise is flawed (and I even voted up that answer), but I disagree with the answer. I genuinely think the Galactic Republic was mostly corrupt. I will approve an answer that convinces me the republic was not corrupt or an answer that shows why the Jedi would support it knowing full well it was corrupt.
Consider these points which show the republic was corrupt and weak:
- The Republic can't break a corporation’s blockade of a defenseless planet in Episode I and many of the senators don't seem to care. If a corporation here on Earth tried to blockade a small country with a private army, would you expect the other nations to tolerate it?
- People are surprised to discover slavery still exists in the Outer Rim worlds. Thus showing the Republic's laws are not enforced, and where a government can't enforce its own laws, corrupt people will take over.
- By Episode 2, thousands of worlds were ready to leave the republic and form the Confederacy of Independent Systems, and I would say many of those worlds saw the republic as corrupt. We know the Sith Lords, Palpatine and Count Dooku, were behind the CIS, but the fact that thousands - rather than mere dozens - of worlds were ready to secede shows how weak the Republic was.
- The Jedi Council allowed Chancellor Palpatine to appoint a personal representative on the Jedi Council. Jedi Master Mace Windu said it was inappropriate but allows it anyway. (This may show the Jedi were more under the thumb of the Republic government than they wished.)
- Watto, the junk dealer on Tattoine, says "Republic credits are no use out here." This shows that if Republic currency was not respected in the Outer Rim, neither were its laws (such as laws against slavery). This is like asking if the US dollar is respected in an outlying state such as Alaska.
- Corporations get their own senators as shown by this line by Chancellor Valorum. "The chair does not recognize the senator from the Trade Federation at this time."
- Articles on Wikipedia, Star Wars Databank, and Wikia describe the Republic as corrupt.
If the Jedi are idealists as they claim, then why serve a corrupt system?
They could have done many things to lessen the corruption.
- Refuse to serve it until the Republic reforms itself.
- Protect people harmed by corrupt officials.
- Use their Jedi powers to expose corruption.
- Arrest the corrupt.
- Only serve republic leaders they believe are not corrupt.
- Serve only local governments they believe are just (e.g. - Naboo or Alderaan), in local ways, but not serve the galaxy-wide government.
- Protect non-corrupt worlds from corrupt organizations.
The Jedi don't do any of these, even though some would be easy to do. (Or "not do" in the case of refusing to serve.) The Jedi are astute. They should have known the extent of corruption. So that means they are serving a system that does not support their values.
Edit to add: I'd imagine many Separatists saw the Republic as corrupt and wanted to free themselves of its corruption. They surely saw themselves as the aggrieved good guys and the corrupt Republic as perpetuating suffering. And to see the nominally good Jedi supporting the corrupt Republic in battle probably harmed the reputation of the Jedi. I think a politically astute Jedi corps would have been aware of how their opponents would perceive them.
Edit to add: Qui-Gon Jinn knew the Republic was corrupt. He shared his thoughts about the corruption with his teacher, Count Dooku, as evidenced by Dooku's own admission. Dooku claims Qui-Gon not only knew about the corruption, but went along with it because Qui-Gon did not know the truth, or so he tells Obi-Wan. The implication is that Qui-Gon knowingly supported a corrupt government. (On the other hand, Dooku might be flavoring the truth about Qui-Gon's views or outright lying to Obi-Wan.)
Dooku: Qui-Gon always spoke very highly of you. I wish he were still alive. I could use his help right now.
Kenobi: Qui-Gon Jinn would never join you.
Dooku: Don't be so sure my young Jedi. You forget he was once my apprentice as you were once his. He knew all about the corruption in the Senate, but never would have gone along with it if he had learned the truth as I have.
That conversation comes from this scene in the movie, Attack of the Clones.
So the point is that some Jedi knew the Senate was corrupt, and yet they went along with it.