Is it the Christian's responsibility to fight "territorial spirits?"
Absolutely!!!
When we witness, teach, preach, pray, etc., we are fighting directly against Satan's kingdom, to deliver souls from darkness to light. That in and of itself is spiritual warfare!
When God sends an evangelist to preach a revival in a certain township/city, that's spiritural warfare against whatever spirit is holding people captive there.
When people pray for revival in a certain place, that's the same thing.
When people come into those services to cause a commotion, what spirit do you think sent them there?
How many heard the stories of witches sent to churches? Who do you think sent them there? The spirit ruling over that area.
So yes, we do fight against territorial spirits.
Somebody already quoted it:
2Cor 10:4
English Standard Version
For the weapons of our warfare are not of the flesh but have divine power to destroy strongholds.
New King James Version
For the weapons of our warfare
are not carnal but mighty in God for pulling down strongholds,
New American Standard Bible
for the weapons of our warfare are not of the flesh, but divinely powerful for the destruction of fortresses.