Incidents like the Crusades, the Inquisition, and the recent priest sex scandal have all been brought forth by the Church's opponents to try to disprove her authority. Why should we listen to corrupt leaders? If they don't act in accordance with the truth, why should we believe that they teach in accordance with the truth?
God used sinful men to write his infallible Scriptures, so to say that God can't use sinful men to infallibly teach His truth through the Church is, in essence, to deny God's ability to use sinful men to write the Bible. The only way you can use corruption to deny Church authority but still acknowledge the infallibility of Scripture is to draw an arbitrary line that cuts off corruption from mere sinfulness. This arbitrary cut-off point, however, is unscriptural.
"Then Jesus said to the crowds and to his disciples, 'The scribes and the Pharisees sit on Moses' seat; therefore, do whatever they teach you and follow it; but do not do as they do, for they do not practice what they teach. They tie up heavy burdens, hard to bear, and lay them on the shoulders of others; but they themselves are unwilling to lift a finger to move them." - Matthew 23:1-4
Here, Jesus tells the people to obey the teachings of the scribes and the Pharisees, even though they didn't follow their own teachings. Jesus did not say, "Don't listen to the Pharisees because they have passed from mere sinfulness to corruption"
"But one of them, Caiaphas, who was high priest that year, said to them, 'You know nothing at all! You do not understand that it is better for you to have one man die for the people than to have the whole nation destroyed.' He did not say this on his own, but being high priest that year he prophesied that Jesus was about to die for the nation, and not for the nation only, but to gather into one the dispersed children of God. So from that day on they planned to put him to death." - John 11:49-53
Again, we see God working through a corrupt man (Caiaphas wanted Jesus killed!) to teach truth (or, in this case, prophesy about the truth).
If someone says that God can't use corrupt leaders to infallibly teach His truth, then they're putting limits on what God can or should do. For one, we can't put limits on God's omnipotence, and to say that He can't work through sinners is to do exactly that. Secondly, we are in even less of a position to determine what God should and shouldn't do. If He wants to use fallible men to infallibly teach His truth, who are we to say he shouldn't? We have no right to say what God can't or shouldn't do.
Here's another thing to think about. At any given time, the Church leadership is, for the most part, just handing down previously defined doctrines. Sure, the Church occasionally defines new doctrines, but that doesn't happen all the time. To reject doctrines merely because the ones handing them down are corrupt is even less of a good reason than to reject them because the ones who defined them were corrupt.
