"This letter was written by a man."
Translation:Ten list został napisany przez mężczyznę.
In this sentence we really need to stress the fact that it was a man, not just a human being who is likely to be a man.
'Ten list był napisany przez człowieka.' sounds like we want to convey that it wasn't written by an alien.
On a side note, even though był is accepted, I recommend został, because it fits the perfective napisany better.