The REALITY is no one really cares on the actual cruise. Well, at least on main stream mass market cruise lines such as Royal Caribbean, Disney, Carnival, Princess, and Norwegian. It's different on upscale luxury cruise lines and they do expect you to dress up, but going to Alaska you'll most likely be booking on one of the mass market cruise lines. Although a majority of cruisers dress up on formal night, there are still a decent number of people that don't dress up fully and are still allowed in the dining room. At minimal, at least wear something decent. If I don't feel like dressing up, I just wear long pants and a dress shirt... like business casual.

With that said, the only time I've really seen the dining room team turn away someone from the main dining room is if they show up dressed up for the pool. The strictest is always been with specialty dining that have a set dress code... for example, on Disney Cruise Lines there are no formal nights but their adult specialty dining venue have some of the strictest dress code around and if you don't follow their dress code for their specialty dining, they'll turn you away. Same goes for invites to dine with the Captain or another officers. However, this isn't something a majority of cruisers have to worry about.