There isn't a single verse that reads "These (fill in the blank) are qualified to baptize".
Rather, we have examples of people being immersed, or of accounts given. These examples give us a basic framework to begin to try and answer the question. But the honest truth is, there isn't enough info to fully reach a conclusion.
Here is what I mean:
In the Gospels, John the Baptist immerses.
Eventually the 12 apostles with Christ leading them, begin to immerse people in the Gospels.
On Pentecost, 3,000 are immersed, but it never states who actually did it.
The 12 Apostles? A likely assumption, but not indicated by the text.
Acts 8, Philip, a deacon, later called Evangelist, begins immersing in Samaria, then immerses an Ethiopian proselyte to the Jewish faith.
Acts 9, Saul is immersed. It reads only "was baptized". Doesn't say who did it. Paul later gives his testimony in
Acts 22, and in verse 16, he makes it sound like he immersed himself. But did he? Did Ananias do it? Someone else in Damascus? We can't say because the account doesn't say.
Acts 10, Cornelius and his house are immersed. Simon Peter calls for it, but it doesn't state he actually did it. We might assume so, but we can't know for certain because no certain verse states who did the immersing.
Acts 13, Saul/Paul and Barnabas are called Prophets and Teachers. The Holy Spirit separates them, and they become missionary evangelists. Later, though, in
Acts 14:14, Luke calls them Apostles.
In
Acts 16, we read that Lydda and her family were immersed, but the text doesn't state who did it. Presumably Paul or Silas, but maybe not?
Later, in the same chapter, the prison warden and his family are immersed. By Paul and Silas? Again, it doesn't say.
In
Acts 18, Paul founds the church in Corinth. We read about many immersions taking place, but the info is vague. However, in
1 Corinthians 1, Paul indicates he personally immersed Crispus and Gaius, but doesn't recall if he immersed anyone else.
Acts 19, Paul seemingly immerses the 12 disciples of John's immersion, but it doesn't specifically state that Paul took them into his arms and dunked them under water; just that "they were baptized".
Finally,
Acts 22, already mentioned above.
After this, there are no other accounts or indications of any immersions.
So, to conclude, what have we?
We have a very special prophet of God, ordained by prophecy out of the OT, immersing Jews in the Jordan river as part of God's will for his work. He even immerses the Christ.
Then, the Christ's disciples begin immersing under His direction. Most likely as training for the New Covenant.
On Pentecost, we can assume with some safety, that the apostles immersed the 3,000. But again, it's only an assumption.
Then we have a deacon who becomes an evangelist immersing folks in
Acts 8.
Next, Paul is immersed, either by Ananias, someone else, or by himself. If by Ananias, let it be said that no particular gift or position is mentioned in regards to him and his ministry.
Then we have the Gentile conversion through Simon Peter's ministry. He probably immersed them, but it doesn't say so specifically.
Next to last, we have a case of two prophets and teachers called to apostleship, who likely performed immersions for Lydda and her family, and for the prison warden and his family, in Philippi.
Finally, we have Apostle Paul immersing at least two men in Corinth, and then, most likely immersing the 12 students he found in the coasts of Ephesus.
So, this is what I see:
I see John immersing people in order to reveal the Messiah to Israel (the Gospels). I see the disciples immersing folks under the direction of the Messiah (The Gospels). Then I see Apostles likely performing immersions (
Acts 2). I definitely see a deacon called to be an evangelist performing immersions, both on a grand and also on an intimate scale (
Acts 8). I see the possibility of a so-called "laymen" immersing one person, while also admitting that such is only a possibility (
Acts 9 with
Acts 22). I see some Gentiles getting immersed, likely by an apostle (
Acts 10). Next, in all likelihood, two missional evangelists later called apostles, immerse a couple of families, one under somewhat normal conditions, the other under very unique conditions (
Acts 16). I see, because of Paul's assertion in
1 Corinthians 1, that he, as an apostle and evangelist (See Greek for "preach the Gospel' in verse 17), immersed at least two men in Corinth, and yet, many others were immersed--something Paul doesn't remember doing in particular--so who performed them (
Acts 18)? Lastly, an apostle immerses 12 men he meets and shares the Gospel with 9
Acts 19).
That leaves us with the following:
One very unique prophet, Apostles in training under the direction of Jesus, trained Apostles, Deacons cum Evangelists, Prophets and Teachers who become Missionary Evangelists, and possibly no ministerial gifts or positions/titles held doing all of the immersing in the writings of the New Covenant.
Is there a take-a-away from all this?
The first thing I notice is that the words "pastor", "elder", and "bishop" are nowhere associated with immersions, even though "deacon" is.
The second thing I notice is that all of the other gifts mentioned in
Ephesians 4:11 are mentioned specifically, and associated with immersions.
This causes me to wonder if we are not doing things quite backwards, where only pastors, elders, and bishops tend to do all of the immersions, while the other four gifts from Christ (along with the deaconate) take a backseat, or passive role in bringing people into New Covenant salvation?
The third thing I notice is that, in order for the five gifts to the church to successfully equip the saints for the work of the ministry, apostles in training, prophets in training, evangelists in training, and teachers in training are going to have to, at some point, begin immersing people in order for them to become fully trained.
The last thing I notice is that since the entire church is commissioned to teach all nations, and to baptize them in the name of the Father, and of the Son, and of the Holy Spirit, then the entire church is ultimately responsible for every immersion. This means that even if there is no "qualified" person to perform an immersion, immersions still needs must take place, or else the entire church cannot successfully achieve its mandate from the Lord.