Dictating to Others
The Apostle Paul, for thirteen of his fourteen New Testament letters (epistles), dictated his words to somebody who wrote them down.
"Apparently the only Epistle that Paul wrote with his own hand was that addressed to the Galatians (Galatians 6:11). He usually dictated his Epistles to a scribe. Paul used Luke quite extensively, along with Silas, Timothy, and perhaps Mark and others as scribes." (Holy Bible, a Faithful Version, page 46).
A person whose job it was to write down what someone else dictates is sometimes referred to by the 17th century word amanuensis. The modern equivalent of this responsibility would be someone who acted as a secretary. Three of Paul's letters state who wrote down the apostle's words.
The grace of the Lord Jesus Christ, and the love of God, and the fellowship of the Holy Spirit be with you all. Amen. The second epistle to the Corinthians, written from Philippi of Macedonia by Titus and Luke (2Corinthians 13:13, HBFV throughout).
Did He Write Anything?
What part, if any, did the Apostle Paul personally write in his letters?
Although Paul (except for Galatians) had someone write down his words, he still chose to handwrite each letter's closing or salutation.

All the brethren salute you. Greet one another with a holy kiss. The salutation of Paul by my own hand (1Corinthians 16:20 - 21).
I, Paul, wrote this with my own hand . . . (Philemon 1:19).
The salutation of Paul by my own hand. Remember my bonds. Grace be with you. Amen (Colossians 4:18, see also 2Thessalonians 3:17 and Galatians 6:11).
When Did He Do It?
When did Paul begin the practice of personally writing only the last or closing part of his letters?
Apostle Paul, at the start of his writing ministry, began his practice of dictating his letters to others to record. The book of 2Thessalonians, written in 51 A.D., testifies of this fact.
The salutation of Paul by my own hand, which is the sign in every epistle - so I write (2Thessalonians 3:17).
Paul's phrase "which is the sign in every epistle," found in his (chronologically) second Biblical book, implies his first writing (1Thessalonians) also had him penning only its closing. 1Thessalonians was written and sent in 50 A.D.
Why Did He Do It?
Why did Paul bother to personally write the last part of his letters?
Paul chose to close his epistles with his own handwriting as it was proof that they were genuine and not a fake. His handwriting used large letters (Galatians 6:11), likely due to his "thorn in the flesh," which was a disease of the eyes (see 2Corinthians 12:1 - 9, Galatians 4:13 - 15). Such writing made his words clearly distinctive from what preceded them in the letter.
The genuineness of his correspondence was critical as counterfeit letters purporting to be from him were noted early in his ministry.
Now we beseech you, brethren, concerning the coming of our Lord Jesus Christ and our gathering together to Him, that you not be quickly shaken in mind, nor be troubled - neither by spirit, nor by word, NOR BY EPISTLE, AS IF FROM US . . . (2Thessalonians 2:1 - 2).
Paul's own handwriting also added a personal touch to his correspondence and conveyed his affection for those he served.
Conclusion
The Apostle Paul chose to have someone transcribe his words as he dictated to them. His eyesight was such that it made writing difficult and certainly time-consuming. It was also much easier and faster to have a trusted friend write down what he wanted to say. He still opted, however, to close his letters with his own handwriting to show they came from him and that he deeply cared for the church.