If money is not a concern, I would go during the high season - which is August. Yes it might be cheaper in June or September, but if I had the extra money - I would go in August.
Reasons? First of all - it is cold on the Alaskan waters even in June or September. A number of years ago, my wife and I took a cruise to Alaska on the old Sun Princess. It was a great cruise - but we almost freezed to death. We sat on the deck hundled under blankets to admire the view. We were the only people outside during most of the cruise.
In addition Skagway was cold and damp even in June.
Second reason - if you take the Alaskan cruise - you will generally leave from either Vancouver or Seattle. I have lived in both cities, and still reside in the Seattle area. They are both wonderful cities to explore. As you know it rains quite frequently in the Northwest. August is generally our driest month here. In addition, the summers are wonderful - 75-80 degrees without the oppressive humidity that makes life so miserable in other parts of the US during the summer.
Thus - my first choice is to cruise to Alaska in August, and my second is July. You will also have a great time in June or September - but if you go during that time - dress warmly!!
We have only done one cruise to Alaska and it was in early May which was the second sailing of the season. Maybe we were lucky but the weather and the water was great for all seven days.
As far as the clothes go, I recommend dressing in layers so as that you can take off or put on clothes as the temperature changes.
The prices are much better at that time also. You will love the Alaskan cruise, I promise.
Woody, You will get lots of answers, but we have gone on two Alaska cruises - June 13 & Sept. 8. Maybe we were just blessed, but we had great weather on both of them. We did have a few sprinkles at Skagway in Sept, but that was all. And yes, it's usually cold at the glaciers but that's to be expected. You'll have fun no matter which time of year you pick.
You know we went in Mid Sept. Yes it was on the cooler side on the top of the ship when we were near the glaciers, but we live in So Calif, and by most east coast people are considered wimps. Like I said it was cool. I was really cold when we took the helicopter onto the glacier...but it was ice...what do you expect. We loved it was the last cruise of the season and we had a great time!