If you don't need the car seat anywhere else....You can take the train from the airport to downtown Seattle and then the free bus to the cruise port. My husband and I did this with our cruise luggage for our honeymoon a few years ago. The train is right in the airport and there were many people doing the same thing that morning.
We babywear and took our 14 month old to Hawaii without a stroller (needed the carseat that trip as we rented cars and drove for several excursions).
Alaska is amazing and most of the towns are completely walkable. If you do a shore excursion, you'll be on a bus or van so won't need the seat. There is a train in one of the towns (Skagway?) - little ones would get a real kick out of that.
You'll have a great time! Alaska is amazing and cruising with kids really isn't hard.