Welcome to cruise-chat!
Alaska weather varies so much that you need to be prepared for everything. From May through September it can be anywhere from the 50's to 80's (yes, I said 80's), and rain or sun, often in the same day, or even in an afternoon. The southern part of Alaska - where Skagway and Juneau are located - is actually a rain forest region, which is amazing to some people since this is associated with tropical places.
So...you need to look at bringing clothes that can be layered. Nothing individually heavy like snow jackets, but things that you can put on and remove as needed. It is not unreasonable to bring short sleeved shirts, but have a windbreaker and sweater on top that you can peel off if the temperature cooperates. I wouldn't dwell on dressy clothing. Alaska is a blue jeans and flannel place. On the ship you will find things pretty much the same. It is mostly a casual cruise although having a nice dress or suit for the formal evenings is suggested.