Welcome to cruise-chat Annie. I just looked at the pictures from our Alaskan cruise two years ago. We were in Skagway on May 22nd, so that will give you an idea of the time frame.
Our outerwear ranged from winter coats, ear muffs, etc. in Glacier Bay, to just long sleeved shirts and/or lightweight jackets in the ports. I bought a flannel lined jacket in Skagway and wore that when it was cooler. We were fortunate to have very warm weather but like Dave said, temperatures vary a lot.
If you don't want to take a winter coat, at least take gloves and ear muffs as it is cold near the glaciers. I also brought long johns and they came in handy. My husband is very warm blooded, but our pictures from College Fjord and Glacier Bay show him in his winter coat, gloves, etc., too. He is a camera nut, though, and spent long periods of time on the balcony.
We are leaving in three weeks for our second Alaskan cruise. We can't wait! It's a beautiful place to visit.