Number one, all examples of ordinations and baptisms were always by an ordained priest or minister. Consider the following examples:
- Moses anointed Aaron as high priest, Leviticus 8:12
- Moses anointed Aaron’s sons as priests, Leviticus 8:30
- Moses anointed Joshua as leader of Israel, Numbers 27:23, and Deuteronomy 34:9
- Samuel anointed Saul as king, 1 Samuel 10:1
- Samuel anointed David as king, Samuel 16:13
- Zadok the priest anointed Solomon as king, 1 Kings 1:39
- Apostles anointed the sick, Mark 6:13
- Yahweh anointed Yahshua, Luke 4:18, Acts 10:38
- Yahshua anointed (healed) the blind man, John 9:6
- Apostles laid their hands (ordained) the seven deacons, Acts 6:6
- Deacon Philip baptized; Apostles Peter and John laid on hands for the Holy Spirit, Acts 8:11, 17
- Peter baptized group of gentiles during his visit with Cornelius, Acts 10:45-46
- Paul baptized and laid his hands on disciples in Ephesus, Acts 19:6
- Timothy was ordained by the body of elders and for the granting of the Spirit, 1 Timothy 4:14
- Elders anoint the sick, James 5:14
While we realize that ordination and baptism are not the same, they do share things in common. They were/are ceremonial practices performed by a man ordained to that office.
Number two, there is not a greater commitment in this life than baptism. Because of this, it only makes sense that a minister be involved in the process, including counseling with the prospective candidate.
So based on the many scriptural examples, along with the gravity and commitment of baptism, only those ordained to the office of a minister have the authority to baptize.