ANSWER: Although the phrase 'reprobate mind' is found only once in the King James Bible, there are related phrases that occur several times. Even if we do not know what the definition of a word or phrase is, we can see from its context clues as to what the Bible means when it uses it. The KJV Bible shows three verses in the New Testament which use the phrase in question or one closely related.
"And even as they did not like to retain God in their knowledge, God gave them over to a reprobate mind, to do those things which are not convenient" (Romans 1:28)
"Now as Jannes and Jambres withstood Moses, so do these also resist the truth: men of corrupt minds, reprobate concerning the faith." (2Timothy 3:8)
"They profess that they know God; but in works they deny him, being abominable, and disobedient, and unto every good work reprobate." (Titus 1:16)
The minds of these people became corrupted concerning the faith. They deny God and even pollute their good works. Those who do not like to retain the Eternal in their knowledge eventually take on a reprobate mind. It sounds as if these people are not living the right way. They do not seem to have a desire to please him. The result of their approach to life is this state.
We can often find more of what a verse means by comparing how other Bible translations render a word or sentence. For example, notice the same verses mentioned above taken from translations other than the KJV.
"Furthermore, since they did not think it worthwhile to retain the knowledge of God, he gave them over to a depraved mind, to do what ought not to be done." (Romans 1:28, NIV)
"Now as Jannes and Jambres resisted Moses, so do these also resist the truth: men of corrupt minds, disapproved concerning the faith;" (2Timothy 3:8, NKJV)
"They profess to know God, but by their deeds they deny Him, being detestable and disobedient and worthless for any good deed." (Titus 1:16, NASB)
The same Greek word translated reprobate in the KJV Bible (adokimos, Strong's Concordance Number #G96) is translated as depraved in the NIV version of Romans 1:28, disapproved in the NKJV version of 2Timothy 3:8 and worthless in the NASB version of Titus 1:16.
What do we know about someone who has this kind of mind? We know from Romans 1:28 that the people under discussion did not like to retain the Lord in their knowledge. As a result, he gave them over to this reprobate and virtually worthless state of reasoning.
The people Paul is discussing in Romans chapter one are godless and wicked men who suppress the truth by their wickedness (Romans 1:18). He states regarding these men.
"For since the creation of the world His invisible attributes are clearly seen, being understood by the things that are made, even His eternal power and Godhead, so that they are without excuse, because, although they knew God, they did not glorify Him as God, nor were thankful, but became futile in their thoughts, and their foolish hearts were darkened.
"Professing to be wise, they became fools, and changed the glory of the incorruptible God into an image made like corruptible man — and birds and four-footed animals and creeping things." (Romans 1:20 - 23, NKJV)
Paul is railing against some vile people who crept into the early church and began to teach other things, other doctrines, other gospels, and were successful at leading many astray. It was a problem right from the start, and it has continued to be a problem down to this very day!
We know some of the early heresies of the first century church. Among the leading heresies was Gnosticism, from "gnosis" (knowledge). These liars taught a special kind of "inside knowledge" that led many away from the truth they had recently learned from the apostles!
Many of the epistles are written as a direct defense of "the faith which was once for all delivered to the saints" (Jude 3, NKJV) against the onslaught of Gnosticism and other heresies. Paul was not pulling any punches when he railed against these enemies of the truth, whether you are reading about it in Romans or any of his other epistles. Paul was very concerned about the danger these snakes could do to the church members. He rightly called them debased and ultimately rejected!