Specifically at the time of the Phantom Menace. In reality, it appears that the are choosing the right course, to help the people of Naboo, but in the end, it is exactly what the Sith wanted them to do. Is there anything they could have done that would have fought against the Sith?
Just to be sure, let's assume knowing who the Sith were wasn't possible, otherwise it would be far too easy...
EDIT:
I meant to say a method to prevent the fall either during the events of Phantom Menace or after.
Palpatine replaces Valorum as Supreme Chancellor as a consequence of the vote of no confidence by Queen Amidala.
The Republic doesn't appear to have a significant standing military, presumably relying on individual worlds to defend themselves in combination with the Jedi Knights.
So what could have happened differently?
The Jedi could have supported Valorum and attempted to persuade Amidala that this wasn't the job of the Republic to intervene directly or at least consider that the Jedi Knights successfully defended the Republic for thousands of years previously.
Subsequently, as pointed out in other answers the Jedi council supported (and even embraced) the militarization of the Republic.
Also, arguably the "mass" intervention of Jedi on Geonosis was unwise, specifically the confrontation between Count Dooku and Mace Windu and the large number of knights in the arena. As a result a number of Jedi were killed in order to save one Jedi Knight and an (unapproved) Padawan with ultimately ironic consequences but also weakening the ranks of the Jedi.
(Not that I necessarily believe all this but the question in of itself is speculative :D)
The Jedi generally abhorred politics. That's why Count Dooku had to leave the Jedi Order to pursue his career in politics (and meet Palpatine/Sidious and fall to the Dark Side).
That said, there actually wasn't much the Jedi could have done to prevent the Clone Wars. They were in the service of the Republic, controlled by Palpatine, and to oppose him while he had his "emergency powers" would have been an overt act of treason giving Palpatine all he needed to justify Order 66. They had no direct voice in the Senate, and their best indirect voice (Dooku) had switched sides (in more ways than one; he was both publicly a Separatist, and privately Palpatine's Sith apprentice).
However, the events in the third movie could likely have been prevented. If Obi-Wan had learned (or consciously believed) that Anakin and Padme were lovers sooner, he could have taken steps to protect them both and keep them away from the Emperor, even if that meant Anakin would have to leave the Order. By the time he knew, Anakin was so far gone he trusted no-one. If Mace Windu had taken steps to get Anakin under control beyond simply ordering him to the Council Chamber before going to kill Palpatine, then he would have at least had a fair fight on his hands. If Anakin had been better trained about the Dark Side by Obi-Wan and been able to recognize Palpatine's deception, he may have rejected it outright.
Even if all these had failed, if Obi-Wan or the Council had had any better advance notice of Palpatine's plan, they could have evacuated the Temple and scattered the Jedi to the winds, out of reach of the clone troops, where they'd have been an effective resistance force. And the very last chance for the Jedi to remain an effective force was for the 150 or so Jedi believed to have survived Order 66 to disappear like Yoda and Obi-Wan did; again, they would have been a resistance force if that, but they would have survived to aid the Rebellion once it took proper shape. As it was, they tried to organize in a toe-to-toe battle with the new Empire, and were wiped out with the exception of Yoda and Obi-Wan, who stayed in hiding, and a few Expanded Universe characters introduced as a novel's or game's plot requires (and then usually killed off by story's end).
Abjure the easier path of using the Clones, when they were an unknown quantity, and morally questionable at best. They would have lost some territory, to be sure, but they could have raised an involved, patriotic army rather than chucking the responsibility to the clones.
This has parallels to the Romans in Gibbon's Decline and Fall of the Roman Empire, whereby an increasingly spoiled citizen class outsources defense of the realm to the provincial yokels, and ultimately falls apart...
Palpatine knew the heart of the times, and knew that this wouldn't happen, especially with him in control.
I think they needed to be a little more concerned about where the clones came from too. They weren't suspicious enough. Although this is more attack of the clones time. The secretness of jedi order was definately a problem too but it may have been necessary to get rid of palpatine.... Eventually.... Balance the force... I'm too tired to think straight but u get the idea
The Jedi overall, as it says in at least one sourcebook, became TOO complacent, too confident in the stability of the republic, in the senate and its system.
Their arrogance also led them to believe that the Sith were destroyed forever. This is understandable because it had been nearly 1000 years since anyone had heard of a sith. Of course they went into hiding by the new method instituted by Darth Bane 1000 years earlier when, instead of focusing on numbers as the Jedi order did, he instituted the "Rule of 2" where there would only be two sith, a Master and an apprentice.
This way the dark side power, instead of being used up by hundreds of thousands of Sith, would be concentrated in only 2 at any one time, thus, it was believd, increasing their own power. Also, a Master would only pick a student capable of killing the Master. If the student was unable to kill their Master, then that student was simply not worthy of being the next Sith Lord. So a sith would keep searching for a student that they sensed had at least equal or greater potential in the force, recruit that student and train them to eventually be able to kill their Sith Lord Master. If the student could do that, then the Sith and the dark side would grow stronger and be one step closer to eventually being able to take revenge on the Jedi for their defeat 1000 years earlier.
Each Sith Lord understood that they were living and training for a time in the future that the Sith would be strong enough to defeat the Jedi. They would use deception, lies, trickery and cunning, along with politics and the force to eventually take over the Republic and destroy the entire Jedi order. THis, they believed, would usher in a 10,000 year reign of the Sith over the galaxy.
Yoda blamed himself for the fall of the Republic because he was too blind, too complacent and too set in his ways to see what was really happening. He let his guard down in regards to the treachery of the Sith or even if they might still exist somewhere.
Even for a Jedi, once you have a psychological blind spot where you think that something is not possible (in this case because the sith are believed to be extinct), even if you are sensing something off, as Yoda seemed to a couple of times when in Palpatines office, you will dismiss it and it just won't click.
The bottom line is that the Jedi let their guard down as a whole and this is why they didn't see that the Republic was diseased from the inside. Just like a tree that make look healthy and strong on the outside, but inside it's rotting and dying.
In my opinion the way the prequel storyline was laid out the Jedi had no real way to prevent the fall of the Republic after phantom menace, because by that time he had already laid out the pieces he needed in order to seize control of the Republic and create the separatist. The best method the Jedi had to survive and continue into the future was to separate from the Republic after the clone intervention at the start of the war, and retreat from the temple to one of their hidden enclaves. By doing this they minimise the public backlash, reinforce their own security, and maintain their traditions into the future
The book knowledge from Plagueis is technically legends not canon but it works fairly well with what was shown in the prequel trilogy and the clone wars show.
Generally speaking the events of TPM, everything from the blockade, the vote of no confidence against Valorum, and the fall of the trade federation were part of Sidious's and Plagueis's plans to destabilize the republic and start a civil war which in the long run would turn the Jedi from symbols of peace to warmongers.
There isn't really 1 thing that the Jedi could have done better at the end of episode 1 since at that point it seemed to them like Naboo was more of an isolated incident. What they should have done is investigate the Sith involvement on Naboo and the trade federation much deeper. If they had realized that Maul was just the apprentice, not the master, they would have treated the sudden appearance of a clone army with more suspicion and then possibly gotten on Sidious' trail and defeated him before the Galaxy got afraid of Jedi from years and years of them being generals in a civil war.
The key to the fall of the republic, order 66 and the rise of the empire was a long and bloody civil war (another reason the GAR were clones (humans) and not droids like the CIS forces. Human casualties, even if they were clones still made the Jedi look worse than a pile of scrap metal). Preventing that civil war (or ending it much faster) would have been a way to avoid the republic falling.
