What does it mean that God gives someone over to his sin? We find this not only in the first chapter of Romans but also in the Old Testament. Jeremiah warned the people of Israel that this was exactly what their punishment would be, that God was not going to forbear with them forever but that there would come a time when he would give them up. There would be a point when he would give them over to their sin.
Early in Genesis, at the time of the Flood, we are warned that the Spirit of God does not strive endlessly with men. God is patient, but his long-suffering is designed to give us time to come to ourselves, to repent, to acknowledge him, and to be restored to fellowship with him. But at the same time, we are warned that that forbearance does not go on forever and that there can come a point in our obstinate refusal to repent and to respond to God when he will say it's too late and will abandon us to our sin, withholding from us his saving grace. That's a very terrifying thing to consider.
The idea of giving a person over to his sin is a significant part of the final chapters of the book of Revelation, in which we read of John's vision of the inner sanctum of heaven and of the last judgment. We're told that those who have responded to Christ receive marvelous benefits, but those who have obstinately endured in their refusal to repent receive judgment at the hands of God. God says, "Let him who is wicked be wicked still." There's kind of a poetic justice here. To the people who want to be wicked and refuse to restrain themselves in their sin, God says, "I'm not going to restrain you anymore. I'm going to take the restraints away. I'll take the leash off, give you your freedom. I'll let you do exactly what you want to do. It'll be to your everlasting destruction; it'll be to your dishonor and to your ultimate dismay, but if that's what you want, I'll give you over to it."
Is this giving over active or passive? It's active in the sense that God acts to do it. God actually does give a person over to that person's own desires. It's passive in that God remains passive toward that person's self-destruction.