We've been in June many times. Once without any park tickets and had a great time.
It's been our experience that rain happens in short heavy bursts. If it passes through BWV, you can still swim in the rain. Unless, of course, if you hear/see lightning and thunder.
Community hall is wonderful for the kids. free movies, coloring, video games. You can also get free movies with your member card. Also, they usually post a weeks worth of craft activities that have very small fees. A few dollars..
You can also rent, somewhat affordably, individual bikes from community hall during open hours. The surrey bikes can be rented on the boardwalk.
The entertainers, starting at 7pm?, are great and the night life on the boardwalk will keep you very occupied. Our kids always prefer to see the performers vs. be in the parks.
Depending on when you are there, BWV does have "Future of Cool" on the Bordwalk for a week in June. (Don't recall which week. But, it's later in the month). Usually, lots of games, prizes, and entertainment on the green. Last year, the boy from hannah montana was there. A few years ago, it was the X games athletes doing stunts.
kids 6-12 can take a hour fishing excursion from BWV. about $30 each?? Also, you can charter a 2 hour fishing pontoon boat at YC/BC which runs about $200-$250.
You can view the epcot fireworks from atlantic hall on one end and from the bridge walkway to IG on the other.
cheap lunch options are few. You can get limited sandwiches at Leaping Horse libations and maybe spoodles for pizza and possibly a hot dog cart. We do go to BC for Hurricane hannahs for beach eats and beaches & cream for burgers/shakes/ice cream..
June, to us, is a great time to be there. Have a great trip.