In The Polemics of Infant Baptism, B.B. Warfield writes:
All Protestants should easily agree that only Christ’s children have a right to the ordinance of baptism. The cleavage in their ranks enters in only when we inquire how the external Church is to hold itself relatively to the recognition of the children of Christ. If we say that its attitude should be as exclusive as possible, and that it must receive as the children of Christ only those whom it is forced to recognize as such, then we shall inevitably narrow the circle of the subjects of baptism to the lowest limits. If, on the other hand, we say that its attitude should be as inclusive as possible, and that it should receive as the children of Christ all whom, in the judgment of charity, it may fairly recognize as such, then we shall naturally widen the circle of the subjects of baptism to far more ample limits. The former represents, broadly speaking, the Puritan idea of the Church, the latter the general Protestant doctrine. It is on the basis of the Puritan conception of the Church that the Baptists are led to exclude infants from baptism. For, if we are to demand anything like demonstrative evidence of actual participation in Christ before we baptize, no infant, who by reason of years is incapable of affording signs of his union with Christ, can be thought a proper subject of the rite.
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One Pilgrim's Progress 