Most of the New Testament was written after the fall of Jerusalem and the destruction of the temple. One thing, among others, that resulted from this was that it was no longer possible to practice traditional Judaism, which involved rituals at the temple. For Jews, this meant the development of what has become the modern Rabbinical form of Judaism (although I think this development began even before this, since even by this time there were significant numbers of Jews living all over the Roman empire). For those Jews who accepted Jesus as the Messiah, it now became possible to make a complete break with Judaism and develop as an independent religion.
Now, up to this time, traditionally minded Jews weren't so happy with these blasphemers who claimed that the Son of God came to earth was crucified by the Romans. They weren't allowed (if the New Testament writings can be trusted on this) to preach in the synagogues, and there was a lot of pressure, even persecution, by the traditional Jews. So it isn't surprising that the "Christians" would start to resent the "Jews", i.e those that still wouldn't accept Christ.
The Gospel of John was written over half a century after the alleged crucifiction. By this time, there was plenty of bad blood between the Christian and Jews. Read John - I have always felt that the anti-Semitism comes out quite clearly there.
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