I go early June every year, always first or second week, for business/pleasure. This year I'll be going June 6-11. We also like to take a second trip (and a third, and a fourth) and we do this in the September/October time frame. My take:
It's been a long time since I went in the latter half of June through early August, but I do believe the crowds are considerably smaller in early June. That said, the crowds are much larger than in September/October.
I would call them "managable." It seems to me we never wait more than 20 minutes for anything. We use fastpass for the majors, and just have a bit of flexibility in the way we approach the parks. We note the early entry days (we stay off site much of the time) and avoid those parks. We also try to avoid going anywhere on Saturday's, as that's an enormouse attendance day at most of the parks.
I find for some reason that Wednesday is the best day for MK. There is really no "bad" day for Epcot. We always hit MGM on a Monday, and that seems to have always worked out.
Generally, there are plenty of people there, but it is managable. I attribute this to many school districts in Florida still being in session, and many families in Florida and around the country, even if they're out of school, take at least a week before hitting the vacation trail. That said, don't expect a ghost town. I find Pleasure Island to be most enjoyable during June.
A warning: count on rain, probably every day (late afternoon to early evening) and plan accordingly. Rain doesn't ruin my day. But I've seen it ruin people's days because they wake up, the sun is shining, there's nothing on the doplar, and they assume they won't need gear. Bad assumption. And assume temperatures will approach a hundred, then be glad when they back off to 90.
But look forward to an early June trip. It's a good time to be in Disney.