The answer is in Covenant Theology. God made many and great promises not just to Abraham, but to Abraham "and his seed" for ever. We read that "Abraham believed God, and it was counted to him for righteousness" (Genesis 15:6; Romans 4:3; Galatians 3:6; James 2:23). God then instituted with Abraham the Abrahamic Covenant, with its sign of Circumcision of the male genital organ. This sign marked off every Hebrew male as belonging to the Lord, and sanctified his genetic offspring as also belonging to the Covenant Nation. In intertestamental times, some Jews tried to have their circumcision surgically reversed or removed; see 1 Maccabees 1:15 and 1 Corinthians 7:18.
The Covenant was specifically with Abraham and Sarah. Whereas Eve was "mother of all living" (Genesis 3:20), Sarah was "mother of nations" (Genesis 17:16). Ishmael, the son of Abraham and his wife's slave-girl Hagar, is specifically excluded (Genesis 21:10; Galatians 4:22-31).
Who are the Covenant people? First Isaac and Jacob/Israel, and the Children of Israel. By physical descent, they are marked out by Circumcision as the Lord's people. As such they were sanctified by the blood of the sacrifices and burnt offerings. The sacrificial system provided a daily offering for sin, and an annual Day of Atonement and other festivals. The sacrificial system was completely efficacious for all believers of the tribes of Israel, and the strangers among them who also believed. God provided for blood sacrifices to atone for sin (Leviticus 17:11). The parents' faith was counted unto their children; compare Isaiah 8:18 and 54:13. As soon as a boy was 12 or 13, he was examined by the rabbis and declared to be "/Bar Mitswah/" - Son of the Law. From then on he was on his own; his salvation depended on his own faith, not his parents'.
Secondly, Jesus Christ the Covenant Person. He is the ultimate Seed of Abraham: "Now to Abraham and his seed were the promises made. He saith not, And to seeds, as of many; but as of one, And to thy seed, which is Christ" (Galatians 3:16). He was the One who fulfilled all the sacrificial types of the Law, and made atonement once and for all on the altar of the Cross. All O.T. sacrifices are redundant, and have ceased.
Thirdly, we Gentiles who believe are the Seed of Abraham by adoption and grace. "There is neither Jew nor Greek, there is neither bond nor free, there is neither male nor female: for ye are all one in Christ Jesus. And if ye are Christ's, then are ye Abraham's seed, and heirs according to the promise" (Galatians 3:28-29). The male/female bit is with reference to circumcision, which obviously can only be applied to the male of the species. The girl got her Covenant status from the nearest male; if unmarried, her father; if married, her husband. But the matriarchal line is also important; all the kings of Judah have their mother's name given as well as their father's. Our Covenant Sign is now unisex Baptism, which supersedes Circumcision (Colossians 2:11-15). Circumcision is now a spiritual, not a physical matter (Romans 2:25-29).
Peter proclaimed on the day of Pentecost: "Repent and be baptised every one of you in the name of Jesus Christ for the remission of sins, and you shall receive the gift of the Holy Ghost. For the promise is unto you, and to your children, and to all that are afar off, even as many as the Lord our God shall call" (Acts 2:38-39).
Paul laid down, in his doctrine of marriage: "For the unbelieving husband is sanctified by the wife, and the unbelieving wife is sanctified by the husband: else were your children unclean; but now are they holy" (1 Corinthians 7:14). The child of even only a single believing parent is considered "holy" until he reaches the years of discretion, and gets saved, or fails to get saved.
God cares for the little children. The heathen who sacrificed their children to Molech were totally condemned. So (I believe) are those who procure abortions for their unborn children. Children matter to God. Jesus died for all the children. Every denomination has a theology of childhood and infancy. We dedicate them, or we baptise them. But when they reach the years if discretion and decide for Christ, then they get baptised as adults (if dedicated before), or Confirmed (if baptised as infants before).
BEN CRICK