The most common argument against paedo-baptism is rather simple. The Bible does not directly command it. An explicit command to baptize infants is certainly not necessary to affirm the doctrine, as I think it is a clear implication of all of the statements about baptism in the New Testament when read together. However, I do see an explicit command in the New Testament to baptize infants. This comes from the famous story of Christ blessing children. Many have declined to use this in defense of infant baptism because they see it as a general principle about children or being childlike, and not a direct reference to the sacrament. Even Edmund Schlink denies that this passage should be used. However, I believe there are many implications in these texts of baptism.
"Then children were brought to him that he might lay his hands on them and pray. The disciples rebuked the people, but Jesus said, "Let the little children come to me and do not hinder them, for to such belongs the kingdom of heaven." And he laid his hands on them and went away." - Matthew 19:13-15
" And they were bringing children to him that he might touch them, and the disciples rebuked them. But when Jesus saw it, he was indignant and said to them, "Let the children come to me; do not hinder them, for to such belongs the kingdom of God. Truly, I say to you, whoever does not receive the kingdom of God like a child shall not enter it." And he took them in his arms and blessed them, laying his hands on them." - Mark 10:13-15
"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. Truly, I say to you, whoever does not receive the kingdom of God like a child shall not enter it." - Luke 18:15-17
In the Matthew and Mark accounts, the word Paidia is used. This can refer to either infants or small children. However, Luke makes it clear by his use of the word Brefe That infants are in mind. The children being "brought" in Matthew 19 may also point to this. This story is important enough for all three synoptic gospels to include it. Thus, it should not be passed over lightly. But what does this story mean?
Option 1: It shows that all Christians should have child-like faith
This is a common interpretation of this story. Jesus is using this as an illustration. He is using infants to show us that we should have the same humble and trusting attitude in our relationship with God. While this is certainly in the text, the words of Jesus using this as an illustration are not even used in Matthew's account. Thus, this cannot be the primary purpose of this story.
Option 2: It teaches that all children will be saved
This has been a common defense for the idea that God will save all infants. Because he states that the kingdom of God belongs to them, Jesus is showing that all who die in infancy will inherit the kingdom of God. Whether or not this is true, it is clearly not the point of this passage. This text speaks of children being brought to Jesus, something which could be hindered. This clearly has nothing to do with the death of infants unless Jesus is saying "do not hinder the infants from dying. Let them die so that they can come to me." This is simply nonsensical.
Option 3: It teaches that children can be brought to Jesus and enter the kingdom
I see this as the most plausible option. Here is why:
In the beginning of the gospels lies the story of John the Baptist. John was appointed by God to preach the kingdom of God which was approaching. How does one prepare to enter this coming kingdom? Through baptism and repentance. The first time in the New Testament the kingdom of God is mentioned is in Matthew 3 in the context of John baptizing. Why would one assume that Matthew was not referring to baptism when referencing entrance into the kingdom of God in chapter 18 if entrance into the kingdom in chapter 3 is referring to the sacrament? Kingdom and baptism are linked.
There are two other references to baptismal terminology in this story as well.
1. the language of "not hindering"
Early baptismal liturgies often contained the question to the believer, "is there anything hindering you from being baptized?" This may point to the fact that these accounts are using language from baptismal liturgy. The question of course is, did this baptismal liturgy exist yet when the gospels were written? I believe it did. View this statement from the book of Acts:
"And as they were going along the road they came to some water, and the eunuch said, "See, here is water! What prevents me from being baptized?" - Acts 8:36
The same word koluo is used as in the synoptic accounts. The similarity of the language leads me to believe that this was most likely already a part of the churches liturgy.
2. the laying on of hands
More importantly, Jesus lays his hands on the children to bless them. In the early church, laying on of hands and baptism were connected.
"On hearing this, they were baptized in the name of the Lord Jesus. And when Paul had laid his hands on them, the Holy Spirit came on them" - Acts 19:5-6
The author of Hebrews even associates the laying on of hands with one of the elementary doctrines of the faith. "Therefore let us leave the elementary doctrine of Christ and go on to maturity, not laying again a foundation of repentance from dead works and of faith toward God, and of instruction about washings, the laying on of hands, the resurrection of the dead, and eternal judgment" - Hebrews 6:1-2 Most likely, washings and the laying on of hands are referring to baptismal practices.
We must remember that the gospels were written to the developing church. They were not mere biographies of Jesus. This church was growing rapidly and baptisms were being performed every day. Many of the readers would have been newly baptized. Hearing this story of Jesus blessing children, laying his hands on them, telling parents not to hinder there children from coming to him, would certainly bring baptism to mind.
If this passage does not refer to baptism, what is it talking about? How can parents "bring their infants to Jesus" if not by baptism? How else can one be embraced by Christ himself and enter into the kingdom of heaven? Is this merely a narrative about how Jesus liked kids?
That this passage refers to baptism is clear when all the options are considered. This text is not necessary for a belief in infant baptism, but it is the only Biblical command which most likely refers directly to infant baptism.