The Alaska cruise season is very short. Usually mid May-mid September. June, July, August will be the peak months. You have the longest days starting around June 20th. Princess and Holland America have been in Alaska the longest and are highly recommended for the Alaska itineraries.
Was in Alaska twice, 2004 and then 2005. 2004 was in July and 2005 was in June. Need to keep in mind that it is a rain forrest up there and the warmest was around the low 70's. The first time we had a couple of days of drizzle and the next had hardly any drizzle, which they said was very unusual. But the drizzle didn't keep us from anything. My husband loved the temp up there, but there was a variety of ways people dressed there. Whatever they were used to. We were there with the longest daylight hours and I thought we would never get any sleep because of the fantastic scenery and it not getting dark. So really either month is fine.
But I would go with Princess and make sure it is a cruise that goes to Glazier Bay and the College Fjords. That was a fantastic sight!! Just gorgeous.
We went home through Anchorage and our flight wasn't till that night. But Princess rented a hotel where you could keep your luggage and answered any questions. They also had buses running so that you would go into town and spend the day and still get to the airport. I thought they took care of us very well there. Our friends were really impressed with the service.
One of our sister sites, AlaskaPortReviews.com is a great place to start. The overview briefly describes the itinerary choices you have...then scroll down and the menu list on the left leads to information on the ports.
I agree with the others that Princess and HAL are the best choices, mainly for their long history in the area.