As stevenl said, Thailand does not have a hurricane/cyclone/typhoon season. You may be thinking of the monsoon season? June is early in the southwest monsoon /rainy season which affects the west coasts. However, in my experience it’s actually a very good time to visit Thailand and it doesn’t actually rain that much in June - you may experience the odd short shower but you should be fine!
I’ve dived from Phuket in late August, in the midst of rainy season with no problems, although I had to be a little flexible with dive days that time of year.
The Andaman side of Thailand definitely has the best diving! As others have said, there is some very good diving accessed from Phuket. Have you also considered Koh Lanta? The excellent Koh Phi Phi, Bida, Koh Haa & Hin Daeng/Hin Muang sites are accessed from there.