The New Testament is a Jewish book written almost entirely by Jewish people. Most of the concepts in the New Testament cannot be understood apart from their background in the Hebrew Bible.
As far as allegations of anti-Semitism go, in the early days of Christianity, there were no Gentile believers. The whole question of whether Jesus was the Messiah was a family affair to be settled by the family of Jewish people. It is in this context that the tone of many passages depicting criticism of this or that segment of the Jewish people must be seen. The ‘harsh’ passages in the New Testament resemble far more the moral exhortations of the prophets than they do the intolerant rhetoric of medieval sermons.
The real question to be dealt with is not, ‘Is the New Testament Jewish?’ but rather, ‘Is it true?’
How can you believe in the New Testament?