The trouble with Mel Gibson's film "The Passion" is not the film itself, but the gospel story on which it's based. The gospel story, which has generated more anti-Semitism than the sum of all the other anti-Semitic writings ever written, created the climate in Christian Europe that led to the Holocaust. Long before the rise of Adolf Hitler, the gospel story about the life and death of Jesus had poisoned the bloodstream of European civilization.
Once again, no one's arguing that Jesus himself was anti-Semitic, and indeed if we had the narrative of the mythical historical Jesus, the Jesus who was a wise and good teacher who never performed any miracles or said anything that would cause him to be executed in the first place, everything would be okay. But we don't have that Jesus, and the only evidence we have for that Jesus is the wishful thinking of those who don't want to deal with the Jesus we do have, the one who said and did all sorts of outrageous things until the religious and political authorities of that day decided they had no choice but to execute him. If he had only stayed executed, we wouldn't have anything to worry about.
But when in doubt, it's easier to say that Jesus himself was really all right, if only it weren't for Paul who so distorted his message. Ignore for the moment that Paul himself never wrote a gospel. Charles Patterson, the author of the article, takes a strongly revisionist view of the gospels anyway, assuming that they were written outside of Judea in the late first century. This is probably wrong. Mark, at least, was most likely written in the middle of the first century in Judea, and is understood to be essentially Peter’s story. However, I’d be careful of taking either view as certain--there’s evidence, but no certainty, for where and when the gospels were written, or even whether they were originally written in Greek or Aramaic. As for by whom they were written, for that there’s much better evidence, but Patterson blithely ignores it, assuming they were written by anti-Semitic Gentile Christians. John and Matthew, in fact, were written by two of Jesus’s apostles, part of the group whom Patterson calls the Nazarenes, the early Jewish Christians from Judea. Mark was a younger witness to the events, although his family probably came from Alexandria. All of them were Jewish. Luke was the only non-Jew to write a gospel, and I think his comes across as less anti-Semitic than, say, John’s. Luke quotes Jesus as saying "Daughters of Jerusalem, stop weeping for Me, but weep for yourselves and for your children. For behold, the days are coming when they will say, 'Blessed are the barren, and the wombs that never bore, and the breasts that never nursed.'" on the way to the cross. John's gospel looks the most anti-Semitic, largely because he often used "the Jews" to mean "the Jewish leaders," but also because he went to great lengths to demonstrate how the claims of Jesus were so unacceptable to Jewish thought.
I think it might be easier to argue that Judaism is anti-Gentile than that Christianity is anti-Semitic. After all, Judaism claims that the Jews are God's Chosen People. While they aren't inherently superior to others, they are better loved. Christianity doesn't dispute this claim, but rather it argues that Jesus's death on the cross made the Kingdom of God open to all. The supposed anti-Semitism of Christianity is really drawn from two internal debates, one among the Jews and one among the Christians. In this, I think Patterson does a much better job of explaining the history. I believe he's wrong about when, where, and by whom the gospels were written, but I will grant that most of the remaining history he gives is correct. Whereas the Jews were debating Jesus (which is the argument that we see in the Gospels themselves), the Christians were debating the role of the Gentiles. Many of the Jewish Christians regarded Gentile believers as second-class citizens in God's kingdom, and the most "anti-Semitic" of Paul's arguments center around his disputes with this attitude. This led to a certain amount of bad blood between Gentile and Jewish believers, and more bad blood between the Jewish Christians and non-Christians. That made for a whole lot of anger separating the Gentile Christians and the non-Christian Jews.
It is a shameful fact that the anger lasted for centuries in the Christian church, long after the exact causes were forgotten. I think to a large degree this was due to a lack of Biblical literacy, or any literacy, among the people of that era, so when they heard of those disputes they couldn't put them into context as the rough internal debates that they were. However, the strongly philo-Semitic evangelical church (see this article on Judeo-Christian values), in the US and elsewhere, gives the lie to the belief that Christianity is somehow inherently anti-Semitic. In fact, I rather suspect that Biblical literacy and anti-Semitism are inversely correlated. Not in all cases and for all people (Luther, sadly, became rabidly anti-Semitic in his later years, possibly due to disappointment with his overly-optimistic philo-Semitism when he first started studying scriptures), but as a general rule.