I live a couple hours away from Key Largo but make several trips each year for a bit of diving. Usually June has pretty good conditions in my experience, but as with anytime of the year, the weather can turn for the worse. Most of the shallow reef diving is done off a boat that is tied to a mooring ball or has dropped an anchor in the sand. Current may be present, but is not really an issue very often. I've only had a couple of shallow dives where I felt I needed to really work to get back to the mooring line. If top side marine conditions get robust, there can sometimes be a bit of surge present. My experience is that the boat captain's talk and if one signals that a shallow site has nasty current, then the captain on the boat you're on will avoid the site.
I don't have my log book handy, but I think late May/early June last year was in the upper 70's if I recall. Enjoy your trip: Key Largo diving is great.