Haven't cruised with either but have done Alaska cruise with Holland America - absolutely first class and has advantage of being American hence able to berth rather than anchor and be ferried by tende.
Ncl informal with freestyle dining which is good,princess more forma.
We did the Princess Alaskan cruise & combined it with a rockies land stay tour (Fairmont 5* hotels throughout)Could not fault the holiday in any way whatsoever - fantastic but it must be said if you are not into wild life and scenery then don't go. There is so much to see and insufficient time to see it in so it is not what one could say was a relaxing holiday. Princess staff could not do enough for us throughout the 2 weeks-very efficient but friendly with it as were the co-travellers.
Cruised to Alaska on the Princess Line, fantastic. Went round the Hawaian Islands on NCL and never again. I was warned but didn't listen.
Cruised from Seattle to Alaska and back to Seattle on Princess (Dawn Princess) Sept 2006 Put my name on the long list who say this cruise is FANTASTIC! Ship's people wonderful as were the people we met in Alaska. Enjoyed it all, especially 'anytime' eating. Could fit in a meal when we were not watching the scenery all around us.
Try Holland America Line, they are fabulous.
We went to Alaska in May this year with NCL everything about the holiday was first class, we would definately choose NCL again .
We went with Princess and it was as formal or informal as you want it to be. We do not like dressing up and so made use of the 24 hour buffet or other eating places. Food was excellent, far better than P&O cruises which was very good as well.