According to Paul, the baptized are clothed with Christ (Galatians 3:27). In baptism, all the baptized are joined in “one” in Christ Jesus. We can read backwards in Galatians 3 to see what that means.
The “one” is the “one seed” of Abraham (3:16). We might follow NT Wright in understanding that phrase corporately (one family of Abraham) or take the view that the one is Christ. I think Wright has the better reading of verse 16. Regardless, verses 16 and 28 are linked by reference to the “one.” Baptism gives a share in the “one seed.”
According to verse 16, the “one seed” is the recipient of the promise to Abraham. And we are told in verse 14 that this promise is the promise of the Spirit. If baptism makes the baptized a part of the one, and the one is the one seed, and the one seed is promised the Spirit, then by making the baptized a member of the one baptism confers a share in the promised Spirit.
We confirm this point by reading forward from verse 28, just one verse: Those who are baptized belong to Christ, are one in Christ Jesus. And those who are belong to Christ are Abraham’s seed and heirs (v. 29). Heirs of what promise? Reading backward one last time, we have to say that the promise of which the baptized are heirs is the promise of the Spirit.
For Paul, to be baptized into Christ is to be baptized as a son of Abraham and a son of God and so to be heir of the Spirit.