ANSWER: First, we need to discuss where this infant baptism ceremony originated before delving into what the Bible says about it. Evidence suggests this ceremony was introduced by church 'fathers' in what would become the Catholics.
The first known reference to the baptism of an infant is found in the writings of Irenaeus, a late 2nd century Bishop who the Universal church considers a saint. Other references to the practice also exist in the writings of Origen (early to middle 3rd century A.D.) and Tertullian (late 2nd to early 3rd century). At the very minimum, the practice of infant baptism was a standard ceremony in the Roman church starting in the 3rd century.
Catholics consider this ceremony performed on an infant to be a sacrament (a religious act that confers grace on the person receiving it) that washes away "original sin" in the case of babies. They, however, are not the only denomination that practices this unbiblical baptism ceremony (in one form or another). Others who perform this service include the Lutherans, Methodists, Anglicans, Presbyterians, and Episcopalians.
There are certain steps that lead to a Biblically valid baptism. First, God must begin to open a person's mind to the truth and start to draw him or her to Jesus Christ (John 6:44). Next, the person must begin to understand what sin is and that they have disobeyed God. They start to have a deep, heartfelt repentance and remorse over their behavior (see Acts 2:37 - 38). God then leads them to begin changing their life by obeying him. The person then finally makes a conscious choice to accept Jesus as their personal Savior.
Baptism is an external sign that a person has repented of sinning against God and has begun to produce fruits showing that they have repented and begun to follow him (Matthew 3:7 - 8). It also shows a person has accepted Jesus' sacrifice for their sins and is willing to commit the rest of their lives to doing God's will. The choice to go under the baptismal waters is a serious commitment that lasts a lifetime and affects a person's future forever. This is clearly something an infant cannot do.
An infant cannot understand the nature of sin, repent, begin to bring fruits of repentance then contemplate the serious decision of turning over their lives to the Eternal (see Romans 6).
The baptism of an infant or a small child, therefore, no matter how sincerely performed, is not endorsed in the Bible. Not only that, there is NOWHERE in the New Testament where it states God opened a baby's mind to his truth so that they could repent and change their (no doubt incredibly brief) lives!
For further studies into this fascinating "infant" subject please read see 1Corinthians 1:13 - 17, 1Corinthians 10:1, 1Corinthians 12:13, Galatians 3:27, Ephesians 4:5, Ephesians 5:26, Colossians 2:12, Hebrews 6:2 and 1Peter 3:18 - 20.
In conclusion, infant baptism is not a practice found in the Bible but rather a long-held tradition of the Catholic Church that many other denominations have also accepted. Though no doubt carried out by sincere parents and others, the ceremony itself makes a mockery out of what God requires in order for a person to receive his Holy Spirit and become a true believer in him.