The Bible does not specifically address this question. When one looks through the baptisms recorded in the Gospels and the Book of Acts, it would seem that all that was needed was to be a disciple of either Jesus or John the Baptist (in the four Gospels) or to be a godly Christian (in the Book of Acts), a “godly” Christian being one who was sharing the good news of salvation through faith in Jesus Christ and His shed blood on the cross. It was as a result of believing this good news that people were then willing or desiring to be baptized.
Here are a few examples of these godly Christians in the Book of Acts: Peter and the apostles baptized large numbers in Acts 2 as they responded to the message about Christ. Later on, Philip, who was at first selected as one to distribute food to widows in the church in Jerusalem, preached the gospel in Samaria and baptized believers there (Acts 6,8). Still later, Paul baptized some of those who trusted in Christ in the course of his missionary work, but apparently also let others do the baptizing instead of him (Acts 16:33; 1 Corinthians 1:10-17.).
The pivotal passage that answers this question indirectly is found in the “Great Commission” passage (Matthew 28:18-20). This passage records Jesus’ command to make disciples of all nations and includes baptizing as part of the process of making those disciples. If this commission is given to all Christians (as is commonly held), then it follows that the authority to baptize is also given to all Christians.
The Epistles never discuss who is to baptize. What is discussed is the meaning behind baptism. In both the book of Acts and the Epistles, the proper understanding of how one is saved (Acts 19:1-5) and the symbolism involved in baptism (Romans 6) seem to be more important than who is doing the baptism.
Based on Matthew 28:18-20, as well as on the silence of the remaining portions of Scripture concerning this issue, it would seem that any true believer has authority from God to baptize, even as he has the authority from God to evangelize and teach all that Christ commanded.