Warfield on Paedobaptism
From Warfield’s essay, “Christian Baptism” (Presbyterian Board of Publication 1920), reprinted in Selected Shorter Writings, Vol. 1, (325-331)
There is nothing in the whole history of the people of God which they value more highly, on which they more deeply felicitate themselves, on which they more securely depend, than that they are called by the name of the Lord. It was to this fact that they appealed when in their affliction they turned to the Hope of Israel, the Savior thereof in time of trouble: "Thou, 0 Jehovah, art in the midst of us, and we are called by thy name: leave us not" (Jer. xiv. 9). It was in this that their jubilation reached its height: "I am called by thy name, 0 Jehovah, God of hosts" (Jer. xv. 16). When our Lord commanded his disciples to baptize those whom in their world-wide mission they should draw to Christ "into the name of the Father and of the Son and of the Holy Spirit," precisely what he bade them do was to call them by the name of the Triune God, that they might be marked out as his and sealed to him as an eternal possession.
Naturally, therefore, this sign and seal belongs only to those who are the Lord's. Or, to put it rather in the positive form, this sign and seal belongs to all those who are the Lord's. There are no distinctions of race or station, sex or age; there is but one prerequisite -- that we are the Lord's. What it means is just this and nothing else: that we are the Lord's. What it pledges is just this and nothing else: that the Lord will keep us as his own. We need not raise the question, then, whether infants are to be baptized. Of course they are, if infants, too, may be the Lord's. Naturally, as with adults, it is only the infants who are the Lord's who are to be baptized; but equally naturally as with adults, all infants that are the Lord's are to be baptized. Being the Lord's they have a right to the sign that they are the Lord's and to the pledge of the Lord's holy keeping. Circumcision, which held the place in the old covenant that baptism holds in the new, was to be given to all infants born within the covenant. Baptism must follow the same rule. This and this only can determine its conference: Is the recipient a child of the covenant, with a right therefore to the sign and seal of the covenant? We cannot withhold the sign and seal of the covenant from those who are of the covenant.
The baptism of infants, no doubt, presupposes that salvation is altogether of the Lord. No infant can be the Lord's unless it is the Lord who makes him such. If salvation waits on anything we can do, no infant can be saved; for there is nothing that an infant can do. In that case no infant can have a right to the sign and seal of salvation. But infants in this do not differ in any way from adults; of all alike it is true that it is only "of God" that they are in Christ Jesus. The purpose of Paul in arguing out the doctrine of signs and seals, was to show once for all from the typical case of Abraham that salvation is always a pure gratuity from God, and signs and seals do not precede it as its procuring cause or condition, but follow it as God's witness to its existence and promise to sustain it. Every time we baptize an infant we bear witness that salvation is from God, that we cannot do any good thing to secure it, that we receive it from his hands as a sheer gift of his grace, and that we all enter the Kingdom of heaven therefore as little children, who do not do, but are done for.
You can read the essay in its entirety here: Warfield: "Christian Baptism"
You might also be interested in Warfield’s response to Baptist theologian A. H. Strong’s case against infant baptism in his widely-read Systematic Theology. Warfield: The Polemics of Infant Baptism