It starts in Genesis and is stated clearly in Genesis 3:15 where Yahweh tells Satan, "I will put enmity between thee and the woman, and between thy seed and her seed." The same Hebrew word zerah, meaning seed or descendant, is used in both places. So the Bible says that Satan is to have just as literal children as Eve. In John 8:44 Yahshua identified these children of Satan as the Jews.
Question: Is it true that the New Testament criticism of the Jews is quite mild?

Answer: The New Testament contains more than just a mild criticism of the Jews. In fact, it is extremely anti-Jewish. It is true that some Christians maintain that the New Testament attack on the Jews is quite mild. Yet, how mild is the "bloody commission," a statement that places upon the Jews guilt for all the righteous blood ever shed upon the earth (Matthew 23:35), or a spurious self-condemnation whereby the Jewish people supposedly take upon themselves and their offspring an eternal blame for the death of Jesus (Matthew 27:25), or a declaration that the Jews are the children of the devil (John 8:44) and that the Jews are a "synagogue of Satan" (Revelation 2:9, 3:9). Paul summed up the denunciation of the Jews with the inflammatory declaration that the Jews are "hostile to all men" (1 Thessalonians 2:15).
The attitude of the New Testament authors toward the Jews is very clear.