I am not sure how quickly the Alaskan cruises fill up. However, as a general rule, you should book as soon as you know what ship and date you want to cruise. The earlier the better your chance of getting your first choice on everything. I do know the Alaskan cruises are popular, and the season is short (about 5 months). For some reason few people cruise Alaska in December.
If it is crowds you want to avoid, you need to look at space ratio. A 100,000 GRT ship with 3,000 people will feel more crowded than a 50,000 GRT ship with only 1,250 people on it.
That said, Carnival uses the Spirit in Alaska. The Spirit class ships have Carnival's best space ratio (around 40). I was on the Pride (younger sister of the Spirit) with 2300 passengers (100% is 2,124) and it never felt crowded.