JOEL - The Bible of course says nothing about him even being baptized. However our Doctrine and Covenants scriptures say:

"Which gospel is the gospel of repentance and of baptism, and the remission of sins, and the law of carnal commandments, which the Lord in his wrath caused to continue with the house of Aaron among the children of Israel until John, whom God raised up, being filled with the Holy Ghost from his mother’s womb.
For he was baptized while he was yet in his childhood, and was ordained by the angel of God at the time he was eight days old unto this power, to overthrow the kingdom of the Jews, and to make straight the way of the Lord before the face of his people, to prepare them for the coming of the Lord, in whose hand is given all power." (D&C 84:27-28)

So we know that he was baptized as a child and received the Holy Ghost in the womb as also recorded in Luke 1:15. We also know that as he grew his parents must have carefully reared him in gospel principles(Luke 1:80), for he came forth from the desert preaching repentance (Matt. 3:1-2).
John's father, Zacharias, was a priest and literal descendant of Aaron (Luke 1:5), and therefore a legal heir to the Aaronic priesthood; so we might speculate that his father was the one who baptized him as a child.
Even though John was also a literal descendant of Aaron and thus holder of the same priesthood, apparently a greater authority was required for his special mission (Luke 1: 68-79), which he received, not from his father, but from an angel of God as described in the scripture above. (D&C 84: 28)

Link to Scripture