I wouldn't say you could blame them directly, but they helped start a series of civil wars that would ultimately end the Republic. I think some of the effects of the Marian Reforms, which gave a consul/general a lot more power when it came to the management of armies, and thus allowed him to (briefly explained) assume dictatorship if the situation allowed, such as Caesar did and Sulla before him.

In the past citizens were called up to the army for specific campaigns but after the Marian Reforms Rome had a proffesional standing army who looked to their commanders for rewards and thus they were loyal to them instead of the state itself. Therefore ambitious generals like Caesar was able to aquire much more power through force than previously.

I believe this was one of the main reasons the Republic fell. Many senators feared Caesar because they feared he was destroying the Republic and thus hindering their own political advancement. As a sidenote when Caesar was assassinated, Octavian managed to transfer the people's love for Caesar to himself, and therefore he had much support when he assumed all the power in Rome.

So in my opinion Cato and the senate were not really directly responsible for the downfall of the Republic, but rather the military system that Rome had in place were the reason. While it certainly could've been avoided if the had Senate chose to react differently, it was really inevitable.

Source: The Fall of Carthage, Antony and Cleopatra; both by Adrian Goldsworthy