June is a great time to sail Alaska. While no one can really predict the weather there is "usually" less rain at this time and the temperatures are good for Alaska. You will find lows in the upper 40's and low 50's F and highs in the 60's and higher.
The key to dressing for Alaska is layering. Start with a light shirt and then add a sweater or sweatshirt and then a water proof jacket. You can add or remove clothing as needed. A good pair of walking shoes is a must. Boots aren't really needed unless you are going on some "major" hiking excursion.
Have a great time! Alaska is beautiful!