The basic arguments for the Reformed view of Baptism including infants note the parallel of circumcision and baptism and the household formulas in Scripture. Here is an explanation:
In ancient covenant procedure, you took an oath of allegiance (i.e. “the pledge to God for a good conscience” of 1 Peter 3:21) and did rites symbolizing the ordeal sanctions of the covenant (enacting what would happen to you if you failed to keep covenant). Whenever someone went to their king in the ancient near east and went through the ancient covenant procedure they did so on their own pledge and their pledge of loyalty to the king was for those that he or she represented. Thus under the Old Covenant, the servant of God would be circumcised and have his children circumcised. And under the New Covenant, the servant of God would be baptized and have his or her children baptized. God deals with families in terms of the representative authority of the head.
The pattern of God’s dealings with humanity is by household – during both the Old and New Testaments. Everyone agrees that this is so during the Old Testament. Infants were circumcised because they belonged to the household of one who chose to follow the Lord. We do not need to establish whether there were infants in the households mentioned in Acts or not. What we need to establish is whether God still works with households in the New Testament just as he did in the Old Testament.
Genesis 7:1 Then the LORD said to Noah, “Go into the ark, you and all your household, for I have seen that you are righteous before me in this generation. (ESV, emphasis added)
Joshua 24:15 And if it is evil in your eyes to serve the LORD, choose this day whom you will serve, whether the gods your fathers served in the region beyond the River, or the gods of the Amorites in whose land you dwell. But as for me and my house, we will serve the LORD.” (ESV, emphasis added)
Those times when the little children are excluded it was expressly stated: i.e., Gen 50:7-8; 1 Sam 1:21-22
Matthew 10:12 As you enter the house, greet it. 13 And if the house is worthy, let your peace come upon it, but if it is not worthy, let your peace return to you. 14 And if anyone will not receive you or listen to your words, shake off the dust from your feet when you leave that house or town. (ESV)
Acts 2:38 And Peter said to them, “Repent and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins, and you will receive the gift of the Holy Spirit. 39 For the promise is for you and for your children and for all who are far off, everyone whom the Lord our God calls to himself.” (ESV, emphasis added)
Acts 11:14 he will declare to you a message by which you will be saved, you and all your household.’ 15 As I began to speak, the Holy Spirit fell on them just as on us at the beginning. 16 And I remembered the word of the Lord, how he said, ‘John baptized with water, but you will be baptized with the Holy Spirit.’ 17 If then God gave the same gift to them as he gave to us when we believed in the Lord Jesus Christ, who was I that I could stand in God’s way?” 18 When they heard these things they fell silent. And they glorified God, saying, “Then to the Gentiles also God has granted repentance that leads to life.” (ESV, emphasis added)
Acts 16:14 One who heard us was a woman named Lydia, from the city of Thyatira, a seller of purple goods, who was a worshiper of God. The Lord opened her heart to pay attention to what was said by Paul. 15 And after she was baptized, and her household as well, she urged us, saying, “If you have judged me to be faithful to the Lord, come to my house and stay.” And she prevailed upon us. (ESV, emphasis added)
Acts 18:8 Crispus, the ruler of the synagogue, believed in the Lord, together with his entire household. And many of the Corinthians hearing Paul believed and were baptized. (ESV, emphasis added)
The baptism of households was the norm in the New Testament. When an authority unit of the home professed faith and was baptized, then the entire household was included under his or her authority. Thus children can be told to obey their parents in the Lord (Eph 6:4).
What about verses like Acts 16:31-32?
And they said, “Believe in the Lord Jesus, and you will be saved, you and your household.” 32 And they spoke the word of the Lord to him and to all who were in his house. (ESV, emphasis added)
Children can be instructed to fear God from a very young age.
2 Timothy 3:15 and how from childhood you have been acquainted with the sacred writings, which are able to make you wise for salvation through faith in Christ Jesus. (ESV, NIV says infancy)
The word translated childhood here is the same as the babe in the womb of Elizabeth (Luke 1:41), and the word babe for the baby Jesus (Luke 2:12), and the word infants in the story where the disciples rebuked the people for bringing their very young children (Luke 18:15). It clearly means infants and very young children.
But take a look at the next 2 verses:
Acts 16:33 “And he took them the same hour of the night and washed their wounds; and he was baptized at once, he and all his family. 34 Then he brought them up into his house and set food before them. And he rejoiced along with his entire household that he had believed in God.” (ESV, emphasis added)
Notice that the verb for who believed is singular – it was because of his faith. This is obscured in many translations, but anyone who knows Greek can see that it is a singular verb. If I remember correctly this argument is made in an article in The Case for Covenantal Infant Baptism edited by Gregg Strawbridge.
At the end of the day, whether one is baptized as an infant or an adult the sacrament has dual sanctions. And “drowning” will come to those who respond to God in unbelief and blessing to those who respond to God in faith. God sets forward his promise in the sacrament. How we respond to that promise does not change God’s action. This blog post relies heavily on the explanation of Lane Tipton of Westminster Theological Seminary.