We went the week of 6/16 last year. The cruise the week before us had high 70's low 80's, our cruise was mostly in the high 50s and low 60's, so it can really be a crap shoot. I would gues you can expect 65 to 70 degree highs with lows in the 50's, but again there is no gaurantees.
Bring a rain proof light jacket if you have one. We ended up buying them for $20 a piece (which was a good price for the coat) at our first stop. It can rain almost everyday off and on (its a rainforrest) but then again it can be sunny the whole trip too. Alaska is a hard cruise to pack for. The ship was comfortable the whole time, so if you have a good coat and maybe a sweater or two just in case you should be fine. I brought shorts but never needed them.
Carnival Cruise Lines:
Fantasy 90, Celebration 95, Victory 05, Conquest 05, Spirit 6/06, Freedom 6/08, Holiday 7/08, Victory 6/09, Valor 9/09, Dream 6/10, Splendor 8/10,Conquest 8/11,Magic 11/11,Triumph 4/12,Spirit 7/12,Freedom 11/12, Triumph 2/13 - The fire cruise, Magic 1/14, Freedom x 2 6/14
NCL Dream October 2006, RCCL Mariner OTS June 2007, Celebrity Silhouette 2013