If You Are A Landlord And A Tenant Has Someone Break In A Door, But Nothing Is Stolen, Who Is Responsible For The Repair Of The Door?

Your question implies that the renter damaged the door. Unless you can prove the tenant or their guest damaged the property it is the landlord's responsibility to maintain the property just as it would be if a storm damaged the roof or a water line burst. If the tenant called the police upon discovery of the broken door then the landlord wouldn't have a leg to stand on in court. The fact that nothing was stolen is irrelevant - possibly the intruder wasn't a thief and was looking to rape or kill someone. Bail enforcement agents and repo men are notorious for breaking doors and not stealing things. They usually don't leave a business card. Prove it was the tenant or fix it at your own expense.

In this case, the tenant is to be held responsible. Because the house is in his use and he is responsible for any damages. It's like rent a car , if you have taken it and its door gets broken than you will repair it not the owner.

The tenant. The Landlord supplies the keys for entry. If the tenant is careless and does not have their key for entry and damages the door for entry, it is the tenants responsibility to pay for a new door. If tenant can not repair, then Landlord can deduct cost from security/damage deposit at end of lease.
Landlord has to send a letter explaining deductions along with the return of the deposit at end of lease. It would be a good idea to take a photo of damage.