Simple, but not easy (or not easy at first...).
Get people to perform acts they know in their guts to be evil - then make sure they do not repent them.
(To make them sin is not at all difficult, they will all do this anyway, sooner or later - they are, after-all, sinners by nature. It is preventing repentance that is the difficult thing - by nature many will repent, so you must prevent what is natural.)
That is all you have to do - once this simple system is in place, then it will become self-corrupting.
To corrupt the world, just ensure that repentance is prevented. Ensure that repentance is portrayed as strictly meaningless, hypocritical, weak, lame, lacking in self-esteem. Just ensure that repentance is seen as blaming, repressive, inegalitarian, racist, sexist.
(Because if you repent an evil then you acknowledge an evil, you implicitly accuse others of an evil - and that is to be judgmental. Thus the ethic transforms, inverts: the only evil is now to repent evil...)
Unrepentant sin is a ratchet-process, because once corrupted people stay corrupted.
Instead of repenting acts which they know in their guts to be evil, they boast of them - they encourage others to do the same. Why not?
So Screwtape says: Sin - go ahead - do it, if you feel like it - if it makes you happy...
(The precise sin doesn't matter, so long as you disgust yourself.)
Repentance? Nothing to repent. Be proud of your decisions (they are yours - they define you).
(It is those who believe in repentance who are the real sinners.)
Thus spake Screwtape.
What is an unrepented sin, after all? - well, the answer is that unrepented sin is fixing one's face away from God. The sin is to turn from God, repentance is a turning back to God. So when there is no repentance you are stuck - living with your back to God.
Once a sinner (and it could be just once! - that's the beauty of it) you stay a sinner, and the corrupter can leave you alone to wreck others by your example and teaching, and can apply his attentions elsewhere.
Not tempted by the same old sins? Don't worry, be patient, fashion will provide a supply of novelties to tempt every soul.
It's so easy for Screwtape now!
In the old days people were washed clean of their sins and able to make a fresh start, again and again! - the tempter was never sure of his prey until the very end.
But now, you do one measly unrepented sin and the tempter has you!
The sin cannot be repented, cannot be forgiven (who is there to forgive it? Only a god can forgive sin, and we know that they don't exist...) and the only recourse is to deny the sin; and if you deny the sin then you promote the sin.
The ratchet of corruption is exponential - just a few proud souls at first.
But each unrepentant sinner permanently joins the ranks of the death eaters, and by example or argument brings another one or two out of the fold.
The growth is exponential.
And pretty soon there is a democratic majority to enforce sin - which cannot be repented.
Such a simple system - so very effective...