I imagine the handle of the bucket and the rope are bloody because when the culprit made their trip to the well, they may have decided to clean up a little.

After what was done to Niles, their hands were bound to be bloody and it'd be too risky for them to their room without taking care of the mess a little.

If they placed their hands on the handle of the bucket, after they pulled it up using the rope, it likely caused the blood to easily transfer from their hands to the handle and the rope.