I do want to join the Brotherhood of Steel, but Nick doesn't like them so I'm afraid he'll leave me or get killed by them, since they don't like synths. I really would like to be able to call in a helicopter from them and have them fry whoever is bothering me.
According to the wikia page for Nick Valentine, there are no consequences (good or bad) for choosing to join the Brotherhood of Steel while he is your companion.
Nick does NOT like it if you join the Institute.
However, according to the affinity chart here, Nick doesn't like if you help the Brotherhood of Steel. Joining should have no negative consequences, but helping them (doing quests for them) will negatively affect your relationship with him.
Note: The Brotherhood of Steel just doesn't like anything that isn't human. Danse insults CVRIE when they interact, but it doesn't affect her affinity, either.