I have never been on an NCL cruise so I can't be of any help concerning the dress on board. I have read that common sense dictates and that "smart casual" seems to fit the bill.
I have taken a CruiseTour to Alaska. The key to dressing properly is to layer your clothing. Also be prepared for rain. What we did was to get a rainsuit (I think the Stearns suits we bought were $49.00 each.) and use the jacket as a jacket and windbreaker when needed. We had sweatshirts which we could wear under them if needed. As it warms during the day you can simply remove layers of clothing. Since you are going in July I think you might be surprised at how warm it will be in Alaska.
There should be plenty of room for storing the clothing you will need. Spend some time packing. Get all of the clothing you think you will need. Then go through it and remove half of it. Seriously!