This has become a fairly popular question due to modern denominational teachings. The question is answered in several passages and in several ways.
First we see it in the use of the passive voice. Active voice in NT Greek as well as in English means that the subject does something. Passive voice, however, means that something is being done to the subject, with the subject being of course, passive.
In Acts 2:38, for example, Peter tells the crowd to repent using the active voice, but then tells them to be baptized using the passive voice. This means that folks were to do the work of repenting themselves, but they were to allow someone else to baptize them. Passive voice is also used in 2:41 when Luke writes that “those who had received his word were baptized”. Of course none were instructed to baptize themselves.
Secondly, in texts describing baptisms, we see those who work at baptizing while those being baptized submit passively. What better example of this than our Lord Jesus when he comes to John to be baptized (Matthew 3:13-15). John even reasons that he should be baptized by Jesus, but Jesus insists that John must baptize him in order to fulfill all righteousness. Even Jesus did not baptize himself.
Thirdly, Paul wrote that in baptism, we are “baptized into Christ”, “baptized into his death”, “buried with Christ” and “raised….. to walk in newness of life”, (Romans 6:3-4).
There is no mistaking the connection between baptism and Jesus’ death, burial and resurrection. We should take note in all of this that Jesus did not crucify himself; someone else did that. Jesus did not bury himself; someone else did that, and Jesus not raise himself; someone else did that.
In submitting to baptism, we follow the example Jesus set for us. We allow someone else to lower us into the water and at the same time allow God to crucify our old self with its sin (Romans 6:6). We submit to this burial in the water, and to being raised from it just as Jesus was buried by friends and raised by his Father. You just can’t miss the profound and poetic beauty of it all.
Finally, we see that the real work is done by God, through our faith in him:
“…..and in Him you were also circumcised with a circumcision made without hands, in the removal of the body of the flesh by the circumcision of Christ; having been buried with Him in baptism, in which you were also raised up with Him through faith in the working of God, who raised Him from the dead”, Colossians 2:11-12