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;
Note 5 at Ro 1:28: The reason they did not like to retain God in their knowledge is because the knowledge of God would have convicted them and restrained them from committing such acts. This is the same motivation behind the actions of those who oppose Christianity so strongly today. People want to sin without anyone convicting them.
Note 6 at Ro 1:28: The Greek word translated "reprobate" is "ADOKIMOS," and it means undiscerning, not distinguishing, and void of judgment. In this text it may be understood as "an abominable mind, a mind to be abhorred by God and man" ("The Hebrew-Greek KeyWord Study Bible" by Dr. Spiros Zodhiates).
This is describing the state of a person who has "passed the point of no return" with God. As the context explains, God has revealed Himself to every person who has ever walked the earth. But there comes a point when God's Spirit will not strive with man any longer (Ge 6:3). When that happens, individuals are hopelessly damned because people cannot come to the Father except the Spirit draws them (see note 8 at Joh 6:44 and note 6 at Ac 24:25).
Therefore, reprobate people are people whom God has abandoned, and there is no hope of salvation for them. Paul applied this term to Christians who had renounced their faith in Christ (2Co 13:5-7, 2Ti 3:8, and Tit 1:16; see note 5 at Ac 5:5).
Some people may fear that they are reprobate because of some sin or blasphemy (see note 1 at Mt 12:31) that they have uttered. However, as these verses describe, reprobate people are past feeling remorse or conviction. If people are repentant over some terrible action, then that itself is proof that the Spirit of God is still drawing them and they are not reprobate. Reprobate people wouldn't care.