In my country, when we sense the indwelling of a demonic power we use Old Testament verses to encourage the death of this enemy. However I find this at odds with New Testament passages which encourage us to pray for those who curse us. What is the best way to fight this spiritual warfare?
It's important to distinguish between the human being and the demon inside the person. When you pray against those who are under demonic influence (e.g., witches, the occult, etc.), you are not opposing the person but rather the demonic power using that person, as Paul says:
In opposing those individuals, you are actually opposing the enemy and his forces at work in the person's life. Should the Lord choose to work through your efforts, He will either repel the individual along with the demon or else cast out the demon freeing the individual. Either solution will stop the persecution.
Until the Lord frees the person from the demon, you will not be in a position to minister to that person with the gospel. As Jesus says in Luke 8:
On the other hand, once the Lord frees a person from demonic possession or influence, then we have opportunity to minister to them, and in that situation we must not hold their prior behavior against them. We turn the other cheek to give opportunity for the kindness of God to lead them to repentance, knowing that we were not waring against them but against the enemy.
So do not pray for the death of the person under persecution, but rather pray for the Lord to expel the demonic influence controlling their life. If God answers that prayer and frees them from the demon, then the church should be prepared to reach out to the person to minister to them with the Gospel. In that way you may stop persecution and win back a soul for the Kingdom. This is exactly the pattern we see in Paul's life in Acts 8.