Well, you can look at the Navigators (schedule of what is offered each day). That will give you a pretty good idea of what kinds of activities are available. Look for a 7-night cruise one since that will be the closest to what DCL will offer in Alaska.
We were on a Bahamas cruise on the Wonder. There was more than enough going on to keep us busy -- we didn't even have time to do all the things we wanted to!
My DH and I don't drink, gamble, or smoke either. DCL was a good fit for us. You can also look for (search online) for Princess Patter -- the same daily schedule for Princess. That can give you an idea of what they offer in the way of entertainment, etc. My DM has taken Princess twice to Alaska, and it didn't look like the right fit for this trip. She's going with us on DCL and is really excited to try it out.
Another thing you can do is to read some of the trip reports here and on cruise critic. I've found that reading a few gives you a good idea of the "feel" of different cruises. Not perfect maybe, but I think it can help.
Good luck with your decision!