We've gone in mid June last several trips. Upcoming trip is June 12-19. It will be hot! The humidity can be tough sometimes. My advice, depending on your familie's ages, is to arrive at each park early, have a nice sit-down lunch (air conditioned), take a break back at the hotel in the afternoon, & then back to the park around dinner time for the evening activities. I might also suggest you consider the Unofficial Guide's touring plans so that you can tour efficiently.
As far as crowds, mid June starts to build to fairly heavy crowds. Just another reason to arrive early & use the touring plans.
These ideas have worked well for us.