No the New Testament does claim children are born with sins:
The New Testament proclamations that Jesus was God's perfect sacrifice for the sins of the world and that Jesus fulfilled the law in this regard are simply blatant misrepresentations of the Old Testament teachings.
One of the key assertions of the New Testament (NT) writers was the concept of "original" sin. This doctrine claims that sin is hereditary and is passed to all humans through Adam.While the Old Testament (OT) indicates man has an inclination to sin, it is does not originate from one person (Adam). The NT verses which establish this doctrine leave no room for anyone to escape this hereditary sin from Adam.
http://agnosticreview.com/counterfeitpriest.htm
BELOW is what is said about children.
Jesus Himself said: “Let the little children come to Me, and do not forbid them; for of such is the kingdom of God” (Mark 10:14, emp. added). Paul declared that none who are unclean can enter into the kingdom of heaven (Ephesians 5:5). Jesus added: “[U]nless you are converted and become as little children, you will by no means enter the kingdom of heaven” (Matthew 18:3, emp. added). If children come to the world with a “fallen human nature and tainted by original sin” (to use the words of the Catechism), why would men have to become as little children, who are also “contaminated” with sin? The Bible is clear: sin is not inherited. No baby has ever been born bearing the guilt of Adam’s sin. No one bears the responsibility for Adam’s sin but Adam himself.
http://www.apologeticspress.org/apcontent.aspx?category=105&article=2697