Every baptism administered, including our children's baptism, is, in one sense, more about the Church that it is about the children. That is because the Church is a living organism. It is a body: God's covenant people. The congregation is the covenant family of God. We have often lost the grasp of Covenant Theology which our forefathers had. When we view the congregation as a collection of individuals, then we are not thinking along the lines of Covenant Theology: we are thinking like Baptists. The congregation is a living organism.
As with all living organisms, there is included some dead matter. For instance, the hair and nails of my body (which is a living organism), though they are not living material, they are nevertheless part of the living organism. So when we view the Church as a living organism, we acknowledge that there are those among us who are not living partakers of covenant life.
Baptism sacramentally reminds us of God's covenant faithfulness. It is vital to remember that baptism is a sacrament. So many people view baptism as pertaining primarily to the children being baptized. But that is not the primary focus of baptism. Baptism is first and foremost a sacrament in the full sense of that word. Baptism, like the Lord's Supper, is the wonderful means ordained by God, whereby it pleases Him to communicate to us the promises of the Gospel. The sacraments are God's ordained way to visibly reaffirm His covenant faithfulness. This means that baptism is administered, first of all for the benefit of the whole congregation. Through the sacrament of baptism God's confirms His covenant to His people. The baptism of the child is a means toward that end.
Our children's baptism is God's affirmation that they are received unto grace in Christ; that they are welcome; that they may take refuge in the Savior. We need to point our children to Christ as early as possible, because that's what their baptism communicates. The very sacrament of baptism itself confirms that what our children need is regeneration in order to be living members of the kingdom of God. This is what is signified by the dipping or sprinkling of water. In baptism, God the Father confirms to His Church as an organic body, and to His children individually, that He makes an eternal covenant of grace with us. Every time the baptismal formula is repeated, the congregation as a whole, as well as the individual members of it, may take comfort from the reaffirmation of God the Father's unbreakable and eternal covenant of grace with them. God's children should never be in doubt whether that relationship with Him is an abiding relationship. As often as baptism is administered, we are reminded of these truths and are sacramentally reassured of God's faithfulness.
Baptism reassures God's children of the efficacy of the atoning work of God the Son. We are reminded over and over through the sacrament of baptism that Christ's work for our sins is finished and that He has paid for them all. We have assurance that we may repair the the cleansing fount again and again. We need not be in doubt whether or not our sins have been forgiven. Baptism sacramentally reminds the Church that she has been washed in the blood of the Lamb.
Baptism also demonstrates the sealing work of the Holy Spirit. The Spirit seals us and dwells in us, assuring us that we will be continually sanctified as members of the body of Christ. This is a tremendous reassurance because we often grieve the Holy Spirit with our sin. We learn in baptism, that although we sin, we cannot sin His presence away. He may withhold His operation in us to chastise us for grieving Him, but He will never depart. Baptism is the reassurance of this truth. God sacramentally reassures His people of the ongoing indwelling presence of His Holy Spirit and that the work of the Spirit cannot possibly fail.
It is so important to grasp that baptism, like the Lord's Supper is a Sacrament, not simply an ordinance. Reducing it to an ordinance removes all the emphasis from God's covenantal faithfulness to His people and place the focus on the act of the individual believer. Baptism is for the benefit of the whole congregation, first of all as a sacrament. It is secondary that it initiates the child into the covenant community. Reversing this order, placing the infant being baptized in focus, reveals a Baptistic mindset regarding baptism. It places the individual over the family to which he belongs. In baptism we enter the covenant community, we do not supersede it. This is, at bottom, the difference between credobaptists and paedobaptists. Credobaptists always put, in our estimation, undue emphasis, on the individual and his/her decision to join with God's people.
The Reformed theologians of generations past, in order to affirm their belief that in baptism our children are received unto grace (not: they are recipients of grace), simply quote Genesis 17:7. Why quote from here rather than from a New Testament passage? Because they took all of Scripture in order to establish the doctrine of baptism. The Old Testament is the great presupposition of the New Testament. There is an organic unity of the two Testaments. The entire Old Testament is the foundation of the practice of infant baptism. So when a Baptist asks you to show him a text in the Bible that demands that infants be baptized, you can say, "That is very simple. The entire Old Testament is my text." The entire Old Testament demands the inclusion of children in the visible manifestation of God's covenant people: the Church of the Lord Jesus Christ. The entire Old Testament bears witness to this text of Genesis. The theme is that God is the God of His people and of their seed. This theme is woven all through the Old Testament. It is the very warp and woof of Old Testament theology. That is why the New Testament opens with genealogies. It is demonstrating to us that God is faithful to His covenant to Abraham, Isaac and Jacob.