Lutherans believe that baptizing the infant children of believers is a valid Scriptural practice for these reasons:
1) In Mark 10:14 and its parallel passages in Matthew and Luke where the disciples try to hold back those who were bringing little children to Jesus for him to bless, he says
Suffer the little children to come unto me, and forbid them not: for of such is the kingdom of God.
The Greek word here for “little children” is paidion, meaning “childling, or infant,” the normal Greek word used in the New Testament for infants. It is, for example, the word used of the infant Christ whom the wise men from the East came to see shortly after his birth, and it is the word used when Herod told his soldiers to search for the Christ child in order to destroy him. Jesus clearly approves of parents bringing their little children to him. He even says of them “of such is the kingdom of God.” Even though this is not in the context of baptism, it is very hard to reconcile such a statement with the exclusion of infants from baptism, which is the means by which believers are brought into the kingdom of God.
2) Peter says in Acts 2:38-39 that God’s promise of the remission of sins in baptism, and the resulting gift of the Holy Spirit, was for everyone in his audience, for their children, for everyone far off, and for as many as the Lord would call:
Then Peter said unto them, Repent, and be baptized every one of you in the name of Jesus Christ for the remission of sins, and ye shall receive the gift of the Holy Ghost. For the promise is unto you, and to your children, and to all that are afar off, even as many as the Lord our God shall call.
3) The households of Lydia (Acts 16:15), of the Philippian jailer (Acts 16:33), and of Stephanas (1 Cor. 1:16) were all baptized.
All of these passages suggest infants of believing parents can and should be baptized, and there is in fact no evidence until the 16th century, when the anabaptists appeared, that infants were excluded from baptism. Written evidence outside the Bible (Tertullian, around A.D. 200) and the earliest archaeological evidence (the catacombs, A.D. 200-300) both show that infant baptism was an accepted practice by the end of the second century.
The anabaptists of the sixteenth century reasoned that Mark 16:16 excludes infants because it says “he that believeth and is baptized shall be saved.” They didn’t think infants could be believers because infants’ intellectual and verbal abilities aren’t developed enough for them to hear and believe the Gospel. Many denominations since that time have adopted that reasoning and therefore exclude infants from baptism, but Lutherans still believe the practice of including infants is the correct one because 1) Scripture shows that infants can indeed be moved by the Holy Spirit to believe, and 2) refusing baptism to infants on the grounds that they aren’t capable of faith changes faith from something God gives to us by his grace to something we produce by our own will out of our own abilities; this change in understanding of what faith is severely damages the believer’s life in Christ because it leads one to trust less in God and more in one’s own strength.
Regarding the first point, Luke 1:15 and verses 39-44 show that even an infant in the womb is capable of believing in Christ and being filled with the Holy Spirit. John the Baptist “was filled with the Holy Spirit even from his mother’s womb” and leaped for joy in Elizabeth’s womb when Mary announced to Elizabeth that she was pregnant with Jesus the Savior. The Holy Spirit worked a believing response (joy) to the proclaimed word of the Savior’s coming in an infant still in the womb.
For he [John the Baptist]...shall be filled with the Holy Ghost, even from his mother’s womb...
And it came to pass, that, when Elisabeth heard the salutation of Mary, the babe leaped in her womb; and Elisabeth was filled with the Holy Ghost: And she spake out with a loud voice, and said, Blessed art thou among women, and blessed is the fruit of thy womb. And whence is this to me, that the mother of my Lord should come to me? For, lo, as soon as the voice of thy salutation sounded in mine ears, the babe leaped in my womb for joy. (Luke 1:15, 41-44 KJV)
Also, in Matthew 18:6 Jesus explicitly refers to “little ones which believe in me.”
Regarding the second point, Ephesians 2:8-9 says
For by grace are ye saved through faith; and that not of yourselves: it is the gift of God: not of works, lest any man should boast.
Faith in Jesus, the Son of God, is a gift of God, given by the Holy Spirit to those incapable of coming to him by their own reasoning or strength. To say instead that faith is a product of the believer’s conscious reasoning, and to restrict baptism to those whose faith is a product of such reasoning, makes faith a human work, not a gift. It is important to understand that even in adult believers faith is not of their own doing, not their own work but a work of God's grace. It is always a work of the Holy Spirit by the Word of God. Because Lutherans believe on the testimony of the Scriptures that God can do that work even in infants, we gladly baptize them and welcome them into the kingdom of God.