Question:
How do we as non-Greek (or Hebrew) literate Christians know which translation of the Bible is the most accurate? There are so many to choose from.

Answer:
Let me give an illustration to begin with from 1 Corinthians 13:8-11: “Charity never faileth; but whether there be prophecies, they shall fail; whether there be tongues, they shall cease; whether there be knowledge, it shall vanish away…. When that which is perfect is come, then that which is in part shall be done away…. When I became a man, I put away childish things” (King James Version—KJV).

In this passage we find four words or phrases—fail, vanish away, done away, and put away—which are all translations from the same Greek word, katargeo. It appears that one of the underlying principles of the KJV was the frequent use of English synonyms when encountering the same Greek word several times in a given passage. Supposedly this increases readability.
When I was taught how to write in high school English class, the same principle was promoted—try not to use the same word twice in a sentence. But while this may be good for literary purposes, it is terrible when translating the Greek and Hebrew texts of the Bible. I believe that the proper understanding of the passage quoted above is greatly hindered by the use of four different English words to translate the same Greek word.
Are there any translations that are more faithful to the original Greek in this passage? There are two that I know of that faithfully give the same English phrase (“done away”) all four times in this passage in 1 Corinthians 13—the New Translation of J. N. Darby (JND) and the New American Standard Bible (NASB). Others, including the New International Version, the New King James Version, Today’s English Version, and the Amplified Bible, follow the KJV tradition in this respect.
In other passages, such as John 21:15-17 where the KJV does not make the distinction between two Greek words—agape and philia—for “love,” the NASB falls down, but the JND faithfully makes the very important distinction. In summary, it is my personal experience that the JND translation, while it may not read as easily as the others, most faithfully and accurately follows the Greek and Hebrew texts.
Finally, I would like to encourage all who read this to obtain and use study Bibles whose words are keyed to concordances, such as Strong’s or Wigram’s which are organized by the Greek and Hebrew words as well as the English ones. Better yet, if you are still in college, try to work in at least introductory courses in classical and New Testament Greek and Old Testament Hebrew. It will be hard work, but most rewarding if you desire to be a serious student of the Bible. If you have no plans for college or are through college, you might consider taking correspondence courses in these subjects.