Does weapon elemental damage help minions?

I suspect it currently doesn't for the most part - it will for spectres/dominated of certain monster types (the ones which are explicitly told they have a type of weapon so they meet skill requirements). It should, in theory, happen for any monster which is visually holding a weapon - which includes summoned skeletons. That might be tricky to do, but I've made a note to look into this in more detail and get things tidied up to the best of my ability.
The problem is most monsters don't need a weapon type so mechanically are unarmed despite visually having a weapon in their hand.

For now, don't count on it.