Who can perform baptisms? Can anyone baptize another person?
What is Baptism?
Every believer is commanded to undergo a “believer’s baptism.” The water resembles the burial of the old self; a sort of “watery grave.” They are submerged into the water and then come back up out of the water, picturing a new birth…being a new creation in Christ (2 Cor 5:17). The Apostle Paul asked the church at Rome, “Do you not know that all of us who have been baptized into Christ Jesus were baptized into his death? We were buried therefore with him by baptism into death, in order that, just as Christ was raised from the dead by the glory of the Father, we too might walk in newness” (Rom 6:3-4). This verse does not say that we cannot be saved unless we are baptized. Rather it says that baptism symbolizes our own death to self; being baptized into Christ (or identified with Christ). This is why Paul says we are “baptized into Christ Jesus.” Does water really have special redemptive properties? Are we to be identified with water? No, we are identified by Christ and it is only the blood of the Lamb of God that saves…not water. To say otherwise is to making it a simple mathematic equation: Jesus + water = salvation. We know that is not so (Eph 2:8-9). 
Apparently John the Baptist saw repentance as a prerequisite for basptism because “when he saw many of the Pharisees and Sadducees coming to his baptism, he said to them, “You brood of vipers! Who warned you to flee from the wrath to come” (Matt 3:7). John preached a message of repentance and the forgiveness of sins. Later, Jesus would say that “the kingdom of God is at hand; repent and believe in the gospel” (Mark 1:15), so repentance is always connected to salvation; more so than baptism is. Baptism is an outward symbol of an inward faith that is made public, so it is a public profession of faith, and we’re commanded to profess this faith (Rom 10:9). It’s never a private matter, but every believer is commanded to be baptized. Matthew 28:18-20 is where Jesus’ disciples receive their marching orders. They are told to “Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit” (Matt 28:19). The word “in” is actually “into,” so they are to make disciples, and baptize them into the name of the Father, the Son, and the Holy Spirit. Now they are identified with God. After being baptized in water, they are to teach the new disciples to observe all that Jesus commanded them (Matt 28:20), part of which is the command to make new disciples, and then these disciples go and make new disciples…baptizing the believers as the “go.”
Who Can Baptize?
It’s not so much a matter of who is qualified to baptize. Jesus never baptized anyone, but Himself was baptized as an act of obedience and fulfillment of ever requirement of the Law. John the Baptist baptized a baptism of repentance, but Jesus added repentance and faith or belief (Mark 1:14-15). John taught that they were to repent, and this is also what Jesus taught. John baptized and Jesus commanded baptism for every believer, so who can baptize? If there are no pastors, or church leaders like an elder or deacon, then there is nothing wrong with another believer who has already been saved (and baptized) to baptize another person. There are always exceptions and since there is no direct command about who among believers cannot baptize, we cannot forbid those who baptize who are not ordained. It is a matter of availability. Many small churches (including millions of house churches) have no pastor or elder or deacon, and there are no churches they can attend. Some cannot attend because there are none…and even owning a Bible could cost you your life. It is best if an older Christian leader baptized the person because they must be sure that they have full repented and trusted in Christ. There must be at least a brief question and answer session before the church prior to the baptism, so the person who is being baptized fully understands the profound meaning of it. They must accept the fact that Jesus is fully God, and yet God and Man; that the Son of God is sinless, and has died for those who trust in Him. And that they cannot save themselves because grace is a free gift of God (Eph 2:8-9). If they don’t grasp these basic tenets of the faith, then they are not ready to be baptized, no matter who does it.
Identified with Christ
In 1 Corinthians 10, the Apostle Paul wrote more about baptism and being identified in Christ, writing, “For I do not want you to be unaware, brothers, that our fathers were all under the cloud, and all passed through the sea, and all were baptized into Moses in the cloud and in the sea, and all ate the same spiritual food” (1 Cor 10:1-3). What does Paul mean that they were baptized into Moses? How is that even possible!? Paul is not saying that they were baptized into a person, but they were baptized into or identified with Moses. Ancient Israel was never identified with the Red Sea; they were identified with Moses. The Greek word for baptize (baptizo) rarely only meant to dunk something in a fluid and get it wet. The Greek word means “to submerge” or “immerse,” but here’s how the Greeks used this word. When a fabric was submerged into water and then a dye or color was added, the fabric was then identified with that color. It was not identified with the water or the dye, but only the color. For example, if a garment was dipped (baptizo) into the water and the color purple was added, then it came out as purple. Then fabric being in its watery grave, goes into the dye a colorless fabric, but comes out as being identified by a brilliant, beautiful color. After this, it is no longer identified as a bland piece of cloth, but a beautifully, bright new color. Now, that’s how people will identify it as.
I believe if there are no church leaders available, and there are no prospects in the near future…and, if the person has truly repented and trusted in Christ and believes the essential doctrines of the Bible, then they can be baptized by just about anyone who has also been baptized and saved. It’s not so matter who baptizes, but that they do get baptize. Whatever means they might need to do, they must be obedient to the command to repent and believe, and then get baptized. But wait…it’s not over. Now you must go make new disciples, teaching them the very same things you were taught, and see that they repent and believe, baptize them into the name of the Father, the Son, and the Holy Spirit (matt 28:19). It’s a command. If you believe and have never been baptized, I ask, “Why?” Obey the Lord. Get baptized. Then “Go”….maybe you can’t go into all the world, but t least you can go next door.
Here is some related reading for you: Does Baptism Save You? A Biblical Analysis 
Resource – Scripture quotations are from The Holy Bible, English Standard Version® (ESV®), Crossway Bibles. (2007). ESV: Study Bible: English standard version. Wheaton, Ill: Crossway Bibles. Used by permission. All rights reserved.