I think the clearest proof that *infant* baptism was a tradition delivered by the Apostles is that the first sign of controversy concerning it was whether to perform it as soon as possible from birth, or whether to wait 8 days as the old law required. This is approximately around 250 AD (see the quote below by St Cyprian of Carthage) . This would mean that the practice had to have been long-standing prior. In fact, if the controversy were a matter of *when* to perform holy baptism on the infant, and not whether to baptize, then the idea of infant baptism must have been instinctive by the 3rd century. This would be similar to the Quartodeciman controversy (the celebration of the pasch) in the 150-180 AD. That the Church was supposed to celebrate the Pasch was instinctive , but when to do so was in debate. In fact, the Quartodeciman debate lasted even until it was settled even more so at the Council of Nicaea 325. No one would deny that the Pascshal celebration wasn’t a well established & universal practice.
Now, since the instincts of a generation are the habits of a former, and the habits of a former are the beliefs (see quote from St. Hippolyus of Rome below) of an even earlier generation, than it stands to reason that the debate on when the perform baptism in 250 AD shows that the earliest Christians, who suffered and died for the handing on of the treasure of the gospel delivered to them by the Apostles, practiced infant baptism. You also have to assess this psychologically. None of the authors who taught infant baptism in the early centuries were innovators. They were staunch conservatives. They were the type who *would debate vehemently on when to perform baptism on the infant (weather immediate or on the 8th day), or when to celebrate the Pasch*, since they were so often paranoid over conserving the ancient traditions . Their being that concerned for conservation demonstrates that the practice of infant baptism was a matter known more powerfully than instinct.
The below taken from Catholic Answers.
“He [Jesus] came to save all through himself; all, I say, who through him are reborn in God: infants, and children, and youths, and old men. Therefore he passed through every age, becoming an infant for infants, sanctifying infants; a child for children, sanctifying those who are of that age . . . [so that] he might be the perfect teacher in all things, perfect not only in respect to the setting forth of truth, perfect also in respect to relative age” (St. Irenaeus Against Heresies 2:22:4 [A.D. 189]).
“Every soul that is born into flesh is soiled by the filth of wickedness and sin. . . . In the Church, baptism is given for the remission of sins, and, according to the usage of the Church, baptism is given even to infants. If there were nothing in infants which required the remission of sins and nothing in them pertinent to forgiveness, the grace of baptism would seem superfluous” (Origen – Homilies on Leviticus 8:3 [A.D. 248]).
“The Church received from the apostles the tradition of giving baptism even to infants. The apostles, to whom were committed the secrets of the divine sacraments, knew there are in everyone innate strains of [original] sin, which must be washed away through water and the Spirit” (Origen – Commentaries on Romans 5:9 [A.D. 248]).
“As to what pertains to the case of infants: You [Fidus] said that they ought not to be baptized within the second or third day after their birth, that the old law of circumcision must be taken into consideration, and that you did not think that one should be baptized and sanctified within the eighth day after his birth. In our council it seemed to us far otherwise. No one agreed to the course which you thought should be taken. Rather, we all judge that the mercy and grace of God ought to be denied to no man born” (Cyprian Letters 64:2 [A.D. 253]).
“If, in the case of the worst sinners and those who formerly sinned much against God, when afterwards they believe, the remission of their sins is granted and no one is held back from baptism and grace, how much more, then, should an infant not be held back, who, having but recently been born, has done no sin, except that, born of the flesh according to Adam, he has contracted the contagion of that old death from his first being born. For this very reason does he [an infant] approach more easily to receive the remission of sins: because the sins forgiven him are not his own but those of another” (Cyprian ibid., 64:5).
“Baptize first the children, and if they can speak for themselves let them do so. Otherwise, let their parents or other relatives speak for them” (Hippolytus of Rome The Apostolic Tradition 21:16 [A.D. 215]).
“Do you have an infant child? Allow sin no opportunity; rather, let the infant be sanctified from childhood. From his most tender age let him be consecrated by the Spirit. Do you fear the seal [of baptism] because of the weakness of nature? Oh, what a pusillanimous mother and of how little faith!” (Gregory of Nazianzen – Oration on Holy Baptism 40:7 [A.D. 388]).
“‘Well enough,’ some will say, ‘for those who ask for baptism, but what do you have to say about those who are still children, and aware neither of loss nor of grace? Shall we baptize them too?’ Certainly [I respond], if there is any pressing danger. Better that they be sanctified unaware, than that they depart unsealed and uninitiated” ( Gregory of Nazianzen – ibid., 40:28).
“You see how many are the benefits of baptism, and some think its heavenly grace consists only in the remission of sins, but we have enumerated ten honors [it bestows]! For this reason we baptize even infants, though they are not defiled by [personal] sins, so that there may be given to them holiness, righteousness, adoption, inheritance, brotherhood with Christ, and that they may be his [Christ’s] members” (John Chrysostm – Baptismal Catecheses in Augustine, Against Julian 1:6:21 [A.D. 388]).
“Item: It seemed good that whenever there were not found reliable witnesses who could testify that without any doubt they [abandoned children] were baptized and when the children themselves were not, on account of their tender age, able to answer concerning the giving of the sacraments to them, all such children should be baptized without scruple, lest a hesitation should deprive them of the cleansing of the sacraments. This was urged by the [North African] legates, our brethren, since they redeem many such [abandoned children] from the barbarians” (Council of Carthage – Canon 7 [A.D. 401]).
“What the universal Church holds, not as instituted [invented] by councils but as something always held, is most correctly believed to have been handed down by apostolic authority. Since others respond for children, so that the celebration of the sacrament may be complete for them, it is certainly availing to them for their consecration, because they themselves are not able to respond” (Augustine On Baptism, Against the Donatists 4:24:31 [A.D. 400]).
“The custom of Mother Church in baptizing infants is certainly not to be scorned, nor is it to be regarded in any way as superfluous, nor is it to be believed that its tradition is anything except apostolic” (Augustine – The Literal Interpretation of Genesis 10:23:39 [A.D. 408]).
“Cyprian was not issuing a new decree but was keeping to the most solid belief of the Church in order to correct some who thought that infants ought not be baptized before the eighth day after their birth. . . . He agreed with certain of his fellow bishops that a child is able to be duly baptized as soon as he is born” ( Augustine Letters 166:8:23 [A.D. 412]).
“By this grace baptized infants too are ingrafted into his [Christ’s] body, infants who certainly are not yet able to imitate anyone. Christ, in whom all are made alive . . . gives also the most hidden grace of his Spirit to believers, grace which he secretly infuses even into infants. . . . It is an excellent thing that the Punic [North African] Christians call baptism salvation and the sacrament of Christ’s Body nothing else than life. Whence does this derive, except from an ancient and, as I suppose, apostolic tradition, by which the churches of Christ hold inherently that without baptism and participation at the table of the Lord it is impossible for any man to attain either to the kingdom of God or to salvation and life eternal? This is the witness of Scripture, too. . . . If anyone wonders why children born of the baptized should themselves be baptized, let him attend briefly to this. . . . The sacrament of baptism is most assuredly the sacrament of regeneration” (Augustine – Forgiveness and the Just Deserts of Sin, and the Baptism of Infants 1:9:10; 1:24:34; 2:27:43 [A.D. 412]).
“[W]hoever says that infants fresh from their mothers’ wombs ought not to be baptized, or say that they are indeed baptized unto the remission of sins, but that they draw nothing of the original sin of Adam, which is expiated in the bath of regeneration . . . let him be anathema [excommunicated]. Since what the apostle [Paul] says, ‘Through one man sin entered into the world, and death through sin, and so passed to all men, in whom all have sinned’ [Rom. 5:12], must not be understood otherwise than the Catholic Church spread everywhere has always understood it. For on account of this rule of faith even infants, who in themselves thus far have not been able to commit any sin, are therefore truly baptized unto the remission of sins, so that that which they have contracted from generation may be cleansed in them by regeneration” (Council of Mileum – Canon 3 [A.D. 416]).
The image above is a detail from “The Baptism” painted by Pietro Longhi in 1755.