In early June you will see highs in the upper 50's and low 60's. Lows will be in the 40's. There is always a chance of rain on the Inside Passage.
However: You may see temps in the 70's.
The best thing is to dress in layers. A shirt, sweatshirt and water resistant/proof jacket are the best things to wear. Jeans or other types of rugged pants are also a good choice. Tennis shoes are fine for regular excursions. If you are going to hike then more rugged footwear is required.
A cheap poncho or rain suit is also a must.
Take care and enjoy the beauty of Alaska,