As to infants, our Standards teach that an infant, one or both of whose parents are believers (WCF 28, section 4) — _i.e._, one or both of whose parents profess faith in Christ and obedience to him (WLC., q. 166)–is to be baptized. A bare outline of the abundnant Scriptural evidence of this truth may be stated as follows:

1. In constituting human nature and ordaining the propagation of infant children from parents, God has in all respects amde the standing of the child while an infant to depend upon that of the parent. The sin of the parent carries away the infant from God; so the faith of the parent brings the infant near to God.
2. Every covenant God has ever formed with mankind has included the child with the parent; — _e.g.,_ the covenants formed with Adam; with Noah, (Genesis 9: 9-17); with Abraham (Genesis 12: 1-3; 17:7); with Israel through Moses (Exodus 20:5; Deuteronomy 29: 10-13); and in the opening sermon of the New Testament dispensation men are exhorted to repent and believe, “_because_ the promise (covenant) is unto you and _to your children_“, etc. (Acts 2:38, 39)
3. The Old Testament Church is the same as the New Testament Christian Church.
   1. Paul says (Galatians 3:8) that the covenant made with Abraham (Genesis 17:7) is the “gospel”; and in the whole Epistle to the Hebrews he shows that the Old Testament ritual was a setting forth of the Person and Work of Christ. See above, under chapter 3.
   2. Faith was the condition of salvation then as well as now. “Abraham believed God, and it was imputed to him for righteousness” (Romans 4:3); so that he was the great typical believers, “the father of all them that believe” (Romans 4:11); and all who believe in Christ “are Abraham’s seed, and heirs according to the promise.” (Galatians 3:29) See also the eleventh chapter of the Epistle to the Hebrews. All the Israelites, even those only “according to the flesh”, professed to believe. And all “true” Israelites did believe. “He is not a Jew, which is one outwardly; neither is that circumcision, which is outward in the flesh: but he is a Jew, which is one inwardly; and circumcision is that of the heart, in the spirit, and not in the letter.” (Romans 2: 28, 29)
   3. Circumcision, precisely in the same sense and to the same extent as Baptism, represented a spiritual grace and bound to a spiritual profession. This is taught in the Old Testament, as witness Deuteronomy 10:16 & 30:6. It was the seal of the Abrahamic covenant, which Paul says is the gospel. (Genesis 3:3; 17: 7, 10; Galatians 3: 8) It was the seal of the righteousness of faith. (Romans 2: 28, 29; 4:11) True circumcision unites to Christ and secures all the benefits of his redemption (Colossians 2: 10, 11). And Baptism has now taken the precise place of Circumcision: “For as many of you have been baptized into Christ have put on Christ … And if you are Christ’s then you are Abraham’s seed, and heirs according to the promise.” (Galatians 3: 27, 29)
   4. This Church is identically the same with the New Testament Church. It has the same foundation; the same condition of membership, faith and obedience; sacraments of the same spiritual significancy and binding force. The ancient prophecies declare that the same old Church is to be enlarged, not changed. (Isaiah 49: 14-23; 40: 1-14) The ancient covenant, which was the fundamental charter of the Church, included “many nations” (Genesis 17:4; Romans 4:17, 18; Galatians 3:8), which was never fulfilled until after the expansion of the Church in the New Testament dispensation. And Paul says that the Jewish Church, instead of being abrogated, remains the same through all change — the Jewish branches being cut off, the Gentile branches being grafted in; and that hereafter the Jews are to be restored, not to a new Church, but “_into their own olive tree_.” (Romans 9: 18-24) See also Ephesians 2: 11-22.
4. Infants were members of the Church under the Old Testament from the beginning, being circumcised upon the faith of their parents. Now, as the Church is the same Church; as the conditions of membership were the same then as now; as Circumcision signified and bound to precisely what Baptism does; and since Baptism has taken precisely the place of Circumcision– it follows that the church membership of the children of professors should be recognized now as it was then, and that they should be baptized. The only ground upon which this conclusion could be obviated would be that Christ in the gospel explicitly turns them out of their ancient birth-right in the Church.
5. On the contrary, Christ and his apostles uniformly, without exception, speak of and treat children on the assumption that they remain in the same church relation they have always occupied. Christ, speaking to Jewish apostles, who had all their lives never heard of any other than the old Paedobaptist Church, into which they had been themselves born and circumcised (and their infant circumcision was the only baptism they ever received), never once warns them that he had changed this relation. On the contrary, he says, “Of such is the kingdom of heaven” (_i.e., new dispensation of the old Church). (Matthew 19:14; Luke 18:16) He commissioned Peter to feed the lambs as well as the sheep of the flock. (John 21: 15-17); and all the apostles to “disciple _all nations_,” by first baptizing and then teaching them. (Matthew 28: 18, 19) If only one of the parents is a Christian, the children are said to be “holy” or “saints;” which is a common designation of church members in the New Testament. (1 Corinthians 7:14) In the old Jewish Church every proselyte from the heathen brought his children into the Church with him. So the Jewish apostles write the brief history of their missionary labors precisely as all modern Paedobaptist missionaries write theirs, and as no Baptist missionary every wrote from the first rise of their denomination. There are only eleven cases of Baptism recorded in the Acts and the Epistles. In the case of two of these, Paul and the Ethiopian eunuch, there were no children to be baptized. Five of these cases were large crowds. After Stephanas was baptized with the crowd among “the many Corinthians,” Paul baptized his household. Also were the households of Lydia, of the jailer, of Crispus, and probably of Cornelius, baptized. Thus in _every case_ in which the household existed it was baptized. The faith of the head of the household is mentioned, but not that of the household itself, except in one case, and that as a general fact. The apostles also address children as members of the Church. Compare Ephesians 1:1 with Ephesians 4:1-3 and Colossians 1:1 with Colossians 3:20.
6. This has been the belief and practice of a vast majority of God’s people from the first. The early Church, in unbroken continuity from the days of the apostles, testify to their custom on this subject. The Greek and Roman, and all branches of the Lutheran and the Reformed Churches, agree in this fundamental point. The Baptist denomination, which opposes the whole Christian world in this matter, is a very modern party, dating from the Anabaptists of Germany in A.D. 1637.>> Our Standards teach that precisely the same requirements are made the condition on the part of the parent of having his child baptized that are made the condition of approach to the Lord’s table. WSC q. 95: “Infants of such as are members of the visible Church are to be baptized.” This is explained, WLC q. 166: “Infants descending from parents, either both, or but one of them, professing faith in Christ;” and WCF 28, section 4: “Infants of one or both believing parents.” In the American Directory for worship, chapter 7, the minister is to require of the parents, among other things, “that they pray with and for (the child); that they set an example of piety and godliness before it; and endeavor by all means of God’s appointment to bring up their child in the nurture and admonition of the Lord.” The American General Assembly in 1794, in answer to an overture on the subject, declared that the above passage in the Directory is to be understood as bringing the parent under _an express engagement_ to do as they’re required by the minister. Some have supposed, since the church-membership of the child follows from that of the parent, that every person who was himself introduced into the Church by Baptism in infancy has an indefeasible right to have his children baptized, whether he professes personal faith in Christ or not. But this is manifestly absurd — (a.) Because all members of the Church have not a right to all privileges of church-membership. Thus baptized members have no right to come to the communion until they make a profession of personal faith. Until they do this they are like citizens under age, with their rights held in suspension, as a just punishment for their refusal to believe. These suspended rights are those of communing and having their children baptized. (b.) A person destitute of personal faith can only commit perjury and sacrilege by making the solemn professions and taking the obligations involved in the baptismal covenant. It is a sin for him to do it, and a sin for the minister to help him do it.

Hodge, Archibald Alexander. Westminster Confession, A Commentary. Banner of Truth Trust, 2013, pages 345-349