It is not debated whether Paul wrote Galatians. It is one of the seven undisputed letters alongside Romans, 1 & 2 Corinthians, Philippians, 1 Thessalonians, and Philemon. Its acceptance is due to its similarity to these other letters in style, theology, and vocabulary and the details concerning Paul’s life (esp. Gal 1:10 – 2:14; 4:8–19; 6:14–16). However, there are two things worth noting concerning the production of the letter. First, it is the only Pauline letter where ‘all the brothers who are with me’ is mentioned in the prescript. ‘Brothers’ here can mean his co-workers (e.g. Ellis), but most likely means all the Christian brothers and sisters at Paul’s point of writing. The letter is likely written from Syrian Antioch if I am right about the date. Otherwise, this would include the Christians in Corinth or Ephesus, if the letter is later.
The brothers and sisters are likely mentioned not because they are co-authors or even co-senders, but they endorse the material in the letter. Thus, all those Christians with Paul at his point of writing agree with his appeal and repudiation of the Judaizers. They stand with Paul in advocating that the only real gospel is a gospel of grace and faith. New Gentile believers are not required to yield to the Judaizers’ demands that the male converts are circumcised and that all new believers live by the expectations of aspects of Jewish law that mark them distinct from the world. When combined with Paul’s testimony that the Jerusalem Church endorse his apostleship and gospel (1:17–19; 2:1–10), the whole church stands behind Paul—Jerusalem and Antioch. As such, this pulls the carpet out from under the feet of the Judaizers who are claiming Jerusalem’s endorsement in their repudiation of Paul and his supposedly deficient gospel. The mention of the brothers likely means that they have heard the letter, and may even have contributed to its production.
The other interesting reference in 6:11 where Paul exclaims, ‘see with what large letters I am writing to you with my own hand.’ These words almost certainly indicate that Paul wrote this verse and/or some portions around it. As such, we can surmise that another Christian acted as his amanuensis (secretary), he dictating the letter to him—someone like Tertius in Rom 16:17. If Paul writes from Antioch around 47, this may be John Mark, who is at this stage an essential member of the Pauline team. Alternatively, Barnabas or Titus may have acted on his behalf (Titus mentioned in 2:3). If he is writing later from Corinth or Ephesus, then Tertius may be involved, or Timothy, Gaius, or any of the other brothers or sisters in those cities.
The writing of letters like those of Paul was probably not quick. Each sentence was likely carefully crafted and the labour of writing it down slow and laborious. One can imagine Paul with a crowd of key Antiochian Christians including Barnabas, Titus, and perhaps John Mark, sitting around hearing him dictate. They may have made suggestions as he wrote.
The upshot is that when the letter reached Galatia, delivered by one of Paul’s Antiochian emissaries, it was a letter which the whole Antiochian Christian community vouched for. The reference to the brothers would have added to the authority of the letter.