The advantages of June: longer days (more time to see the scenery), warmer weather. Prices will be higher in June than in May or Sept. We had heard May had the best weather; went the earliest we could (May 5-15 or so) and found lots of weather variation, temps from 40's to 60's (daytime highs), rain where expected and a couple gloriously sunny days.
If you are prepared with the right gear, and willing to enjoy the incredible beauty of Alaska no matter the weather, there is no bad time to go. Enjoy!