How do we reconcile Matthew 28:19 and 1 Corinthians 1:17 on Paul's baptizing people?
This can be best explained by Paul's words in Romans 12:4-5
4 For as in one body we have many members, and the members do not all have the same function, 5 so we, though many, are one body in Christ, and individually members one of another.
If we take another look at the Great Commission in its entirety we see it is a corporate command to a group so that the group as a whole is commanded to perform these functions:
18 And Jesus came and said to them, “All authority in heaven and on earth has been given to me. 19 Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, 20 teaching them to observe all that I have commanded you[plural]. And behold, I am with you always, to the end of the age.” -Matthew 28:18-20
So, with respect to baptism, what is an essential function of the whole, the body, is not the essential function of an individual member, Paul.
Now we know how essential baptism is to Paul:
- For, to Paul, the number of people who have been baptized into the Christ are the same as those who have put on Christ and are children of God:
25 But now that faith has come, we are no longer under a guardian, 26 for in Christ Jesus you are all sons of God, through faith. 27 For as many of you as were baptized into Christ have put on Christ. -Galatians 3:25-27
- For Paul, being resurrected with Christ was conditioned (if statement) upon us being baptized into him:
3 Do you not know that all of us who have been baptized into Christ Jesus were baptized into his death? ...5 For if we have been united with him in a death like his, we shall certainly be united with him in a resurrection like his. -Romans 6:3-5
- The very fact that Paul describes people as being baptized into Christ demonstrates his understanding that Baptism is how we are put into a covenantal relationship with Christ.
This is consistent with the great commission itself which describes baptism as one of two main actions required to disciple nations in his name and the name of the Father and the name of the Holy Spirit - the other being to teach them to observe all of Jesus's commands.
We should also consider that in the context of 1 Cor 1:10-17, Paul is concerned with people divided by which evangelist they follow:
11 For it has been reported to me by Chloe's people that there is quarreling among you, my brothers. 12 What I mean is that each one of you says, “I follow Paul,” or “I follow Apollos,” or “I follow Cephas,” or “I follow Christ.” 13 Is Christ divided? Was Paul crucified for you? Or were you baptized in the name of Paul? - 1 Corinthians 1:11-13
Paul is exacerbated by the division and emotes that he is glad he didn't baptize any of these crazy heads trying to divide the church in his name or Apollos' name or Cephas's name - wanting nothing to do with their nonsense.
14 I thank God that I baptized none of you except Crispus and Gaius, 15 so that no one may say that you were baptized in my name. 16 (I did baptize also the household of Stephanas. Beyond that, I do not know whether I baptized anyone else.) -1 Corinthians 1:14-16
But in his emotional rant, he recalls he did actually baptize people and he's not sure how many.
So when we come to 1 Corinthians 1:17, Paul expresses how he sees his primary function is to preach the gospel (for which he is uniquely qualified and commissioned) and not to baptize. It's not that he isn't supposed to baptize, but it's not necessary that he himself does it. And in this case, the less he does it, the better because he doesn't want his personality to get in the way of Christ. Indeed he doesn't even desire the eloquence of his words to get in the way of the message.
17 For Christ did not send me to baptize but to preach the gospel, and not with words of eloquent wisdom, lest the cross of Christ be emptied of its power. 1 Corinthians 1:17
This may also explain why Jesus did not baptize, despite his baptism commission, as Tony Chan rightly pointed out. He didn't want to inadvertently cause division amongst his many disciples based upon who was personally baptized by Jesus.