We've done early June several times now and we really enjoy it. The crowds are manageable, especially with fastpasses. You do get a shower nearly everyday, but usually they only last a few minutes. The first weekend in June is usually Gaydays, if that makes a difference to you. This just isn't an issue for us, other than heavier crowds during the special event days, it didn't seem any different than any other time of year. Have a great time!!!