Having been to Alaska 3 times, I really don't remember any toddlers/babies on board. Older children, yes, but no babies. There really is nothing for them to do or see in Alaska whereas in the Caribbean you can always take them to the beach.
I really don't know what you will do with her all day except play with her. Personally I would try and get a grandparent or an aunt to look after her while you and DH enjoy yourselves on the cruise. There is so much to see that a little one wouldn't be interested in, like Glaciers, whale watching, salmon fishing, touring. Alaska is wonderful.
I know you would miss her, but it's only for a week and I am sure you both could enjoy some time by yourselves.