There are many threads here on hotels. You may think they are outdated because the cruise season in Alaska has just started within the last week, so there are no 2023 experiences yet. And, there are two ports...Pier 66, which is in the downtown, and has several hotel near by, and Pier 91, which is outside of the downtown, and doesn't really have much close. In either case, the true "downtown" is a fairly compact area, and Uber and taxi rides are very short and inexpensive.
And yes, downtown can be expensive, particularly during cruise season. Only you can decide if you need to stay by the airport. I wouldn't recommend it, but it is an option to meet budgetary needs. Also frequently discussed here.