Why would you want to study a particular version (Koine) of Greek that no one speaks anymore?

The straightforward answer is because it is the language in which the New Testament (NT) was written. In addition, there was written a Koine Greek translation of the Old Testament known as the Septuagint.

Koine (sounds like coin-ay) effectively means common so we have “common” Greek and this was the common tongue in the time of Jesus and the Apostles as well as beyond. Any time you study languages, you will come across the term lingua franca which is used to refer to the common language of a time. For NT times, this was Greek.

1. With the above in mind, you have to study Koine Greek if you want to be able to read the NT in its original language and grasp a deeper understanding of the text.
2. What was written in Greek may not have a direct counterpart in English. This is a great reason for the different translation approaches used between the different English translations of the Bible. Read the Greek to get to the source.
3. The culture in which the original text written in Greek is different from our own present-day culture. This is important for understanding difficult texts that our present-day culture hates or is confused about. 1 Timothy 2:9-15 is a great example. People have often responded to such a passage by siding with worldly culture and thereby rejecting the Scripture (at least on the target passage), or you get those who read it without seeking to understand the full context and thereby conclude, improperly, to take on abusive, error-filled practices. This point also serves to re-emphasize point 2.
4. Revival. Historically, the early church did all its worship in Greek. This became a problem as the western church and the eastern church grew further and further apart. Eventually, the west broke entirely and did things in Latin and the people largely spoke their own native tongue at this point. This brought about a period of spiritual darkness that stuck around until the Reformation. We are in danger of the same sort of spiritual darkening if we fail to continue to seek out the Greek, the original text of the NT Scriptures. Thankfully, we do have many good English translations today, but we wouldn’t have had them without the Greek; if we forget the Greek, we can endanger ourselves to those who would push forward altered translations of the Bible.
5. For the one studying Greek (or any language for that matter), their minds become sharpened. As you learn Koine Greek, you come to understand the Scriptures as those did in the times that it was written and beyond. You also become sharper at noticing key details in the text that can have profound implications to its interpretation. This helps you understand what was originally being said in a text–especially when it may look like something contradictory is being said upon comparing a given text to another area of Scripture. Such a scenario speaks to the difficulties of translation and emphasizes the benefit of understanding the original language in which it was written.

I’m sure I could make more points but already you can see how each point made easily feeds into the others. Also, many of these reasons to study Greek would apply to study Hebrew which is the original language of the Old Testament Scriptures. Sure, you could just stick to the Greek Septuagint but that work is a translation of the original Hebrew. Once again, it is good and profitable to get to the original language.

Now, with all that said, I am not trying to say that every Christian must learn Greek, Hebrew, Aramaic or whatever other languages. I would highly recommend it though. Between Greek and Hebrew, most English speakers will find Greek relatively easier to learn as there are clear similarities between the two languages.

Latin is another language that can prove useful to studying texts from the early church times and beyond.