In a separate thread, I will start to look at some of the unfair things that are said about Benedict XVI in particular, but here I want to defend the institution of the papacy itself. The most definitive and exemplary defence was made by John Henry Newman in his Letter to the Duke of Norfolk, which you can read in full by following the link. But as I see it, the essential points are these, as succinctly as possible.
Assuming Jesus Christ is both God and Man, and that he intended to found a Church (a set of claims that will be defended in the "Questioning Faith" section), it follows that with his divine power he will have protected her from falling into any error that would destroy her teaching authority through self-contradiction, or corrupt the sacraments on which her life depends. She must continue both to teach salvation, and to make it available to man through the tangible symbols of grace.
In that sense, she is infallible, she cannot fail us, and to the degree that crucial decisions at times must come down to the choice of one man (for a group of men may disagree), that man must be the legitimately appointed successor of Peter, the representative and spokesman of the Apostles acccording to Scripture. He will by virtue of his office be protected by the same protection that God extends to his Church as a whole, in those decisions by which he decides the course of the Church. But he will not be so protected in making any decision that is of lesser moment or importance.
From this it follows that, as Newman insists, a Pope is not infallible and may make mistakes of judgement in administrative and scientific matters, as well as personal ones. He may be morally corrupt, he may be philosophically inept, he may be theologically misguided, he may be politically incompetent - but none of this will prevent him from making the right decision if he is forced to make one that will commit the Church on a matter where the integrity of a fundamental doctrine hangs in the balance. Of course there is considerable room for us to argue over whether a given decision does in fact affect such a fundamental doctrine. (The Pope can remove all ambiguity by framing it as an "ex cathedra definition", but this is rare.)
It is clear that much less is claimed for the papal office than most people think. As the First Vatican Council (1870), which was the first to define the doctrine of papal infallibility, makes clear, the Pope shares in that authority and that protection to which the Church herself is entitled, and no more than that. But that is remarkably little, because God allows his creatures - even his prelates - the maximum freedom he can, short of allowing them utterly to destroy his work.
The immense prestige and respect with which the office of Pope is surrounded, though appropriate in view of the role of the Church in our salvation, can be said to accrue to the Pope in person merely by association. The man who holds the office rarely if ever deserves the respect with which the office is invested, and often in history has demonstrated himself to be profoundly unworthy.
It is against this background that one may try to defend the actions and decisions and character of a particular Pope - not on the assumption that as Pope he must be irreproachable or that all accusations against him must be false a priori. As a Catholic, one would like to find him innocent, but never at the cost of truth itself.