“Now they were bringing even infants to him that he might touch them; and when the disciples saw it, they rebuked them. But Jesus called them to him, saying, "Let the children come to me, and do not hinder them; for to such belongs the kingdom of God.”
Short answer
The Church agrees that baptism is never given without faith. Infants are baptized in the faith of the Church, professed for them by their parents and sponsors, because baptism is a free gift of grace. The practice is attested from the second century, and St. Augustine calls it a custom handed down by apostolic authority. A baptized child must come to believe for himself as he grows.
Drafted from primary sources. Every quotation is checked against the page it links to.
Where this teaching appears on the Offshoots timelines: 1527, The Anabaptists: Michael Sattler on trial; 1530, Baptism for believers only, as the Lutherans recorded it.
15 quotations in date order
“Repent, and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins; and you shall receive the gift of the Holy Spirit. For the promise is to you and to your children and to all that are far off, every one whom the Lord our God calls to him.”
“And when she was baptized, with her household, she besought us”
“In him also you were circumcised with a circumcision made without hands, by putting off the body of flesh in the circumcision of Christ; and you were buried with him in baptism, in which you were also raised with him through faith in the working of God, who raised him from the dead.”
“For He came to save all through means of Himself — all, I say, who through Him are born again to God — infants, and children, and boys, and youths, and old men. He therefore passed through every age, becoming an infant for infants, thus sanctifying infants”
“And so, according to the circumstances and disposition, and even age, of each individual, the delay of baptism is preferable; principally, however, in the case of little children. … The Lord does indeed say, Forbid them not to come unto me. Let them come, then, while they are growing up; let them come while they are learning, while they are learning whither to come; let them become Christians when they have become able to know Christ.”
“But in respect of the case of the infants, which you say ought not to be baptized within the second or third day after their birth, and that the law of ancient circumcision should be regarded, so that you think that one who is just born should not be baptized and sanctified within the eighth day, we all thought very differently in our council. For in this course which you thought was to be taken, no one agreed; but we all rather judge that the mercy and grace of God is not to be refused to any one born of man.”
“Be it so, some will say, in the case of those who ask for Baptism; what have you to say about those who are still children, and conscious neither of the loss nor of the grace? Are we to baptize them too? Certainly, if any danger presses. For it is better that they should be unconsciously sanctified than that they should depart unsealed and uninitiated. A proof of this is found in the Circumcision on the eighth day, which was a sort of typical seal, and was conferred on children before they had the use of reason.”
“And if any one seek for divine authority in this matter, though what is held by the whole Church, and that not as instituted by Councils, but as a matter of invariable custom, is rightly held to have been handed down by apostolic authority, still we can form a true conjecture of the value of the sacrament of baptism in the case of infants, from the parallel of circumcision, which was received by God's earlier people”
“If they must not be called penitents because they have not the sense of repenting, neither must they be called believers, because they likewise have not the sense of believing. But if they are rightly called believers, because they in a certain sense profess faith by the words of their parents, why are they not also held to be before that penitents when they are shown to renounce the devil and this world by the profession again of the same parents? … But yet who knows not that the baptized infant fails to be benefited from what he received as a little child, if on coming to years of reason he fails to believe and to abstain from unlawful desires?”
“Likewise it seemed good that whosoever denies that infants newly from their mother's wombs should be baptized, or says that baptism is for remission of sins, but that they derive from Adam no original sin, which needs to be removed by the laver of regeneration … let him be anathema. … For on account of this rule of faith (regulam fidei) even infants, who could have committed as yet no sin themselves, therefore are truly baptized for the remission of sins, in order that what in them is the result of generation may be cleansed by regeneration.”
“Just as a child, when he is being baptized, believes not by himself but by others, so is he examined not by himself but through others, and these in answer confess the Church's faith in the child's stead, who is aggregated to this faith by the sacrament of faith.”
“If any one saith, that little children, for that they have not actual faith, are not, after having received baptism, to be reckoned amongst the faithful; and that, for this cause, they are to be rebaptized when they have attained to years of discretion; … let him be anathema.”
“It is beyond doubt that the preaching of the Apostles was normally directed to adults, and the first to be baptized were people converted to the Christian Faith. … However, as was mentioned above, the practice of baptizing children rests on an immemorial tradition originating from the Apostles, the importance of which cannot be ignored; besides, Baptism is never administered without faith: in the case of infants, it is the faith of the Church.”
“Born with a fallen human nature and tainted by original sin, children also have need of the new birth in Baptism … The sheer gratuitousness of the grace of salvation is particularly manifest in infant Baptism. … The practice of infant Baptism is an immemorial tradition of the Church. There is explicit testimony to this practice from the second century on, and it is quite possible that, from the beginning of the apostolic preaching, when whole "households" received baptism, infants may also have been baptized.”