The baptism in 1 Corinthians 12:13 is water baptism. The "body" is the local church (in this case, the church at Corinth). The word "by" is the same as in Luke 2:27, "And he came by the Spirit into the temple...", meaning he was led by the Spirit into the temple. The greek word εν is usually translated in or on with the locative case and by (means of) in the instrumental case, as it is here.
Paul's illustration of the "body" in 1 Corinthians 12 is an object, not an individual, nor an organism. He's teaching that every member of the local church is necessary to the functioning of the body as a whole. The same Spirit has led all of the many members into this relationship, each one having been baptized (in water) because it is a prerequisite to church membership.
All being made to drink into one Spirit is consistent with what Peter told them on the day of Pentecost (Acts 2:38):
"Then Peter said unto them,
Repent, [Repentance and faith in Christ -- 1st pre-requisite]
and be baptized every one of you in the name of Jesus Christ for the remission of sins, [Baptism in water -- 2nd pre-requistie]
and ye shall receive the gift of the Holy Ghost." [Church membership -- where the Spirit resides as Administrator of Truth, Comforter, etc.]
Paul stated in Ephesians 4:5, "One Lord, one faith, one baptism", and that's water baptism.