Need advice for an Alaskan Cruise on Princess

Casual dressing doesn't present a problem for Princess in any of their dining rooms or Specialty restaurants.

Many people just dress nicely & can get by even with less than what you describe.

Since you mentioned this likely being your only trip to Alaska, please consider adding a land portion that includes Denali National Park. You can do this with Princess, or yourself. The southeast arm that the round trips include is great, but visiting the main part of Alaska is another realm.

Good to know that you are planning to book the Ruby Princess with a mini suite cabin.

Hope you and your family enjoy the cruise !