June is a good time to cruise Alaska for a couple of reasons. One, it's before peak season and prices are little cheaper. Two, it's near the summer solstice. It kind of cool to take a picture of your spouse at 4am without needing a flash. Three, although weather is not predictable, historically speaking, June is dryer.
As for cruiseline. Princess and Holland America have been doing it the longest, have the most infrastructure in place, and have great itineraries (land and sea). They also sell out very fast. At minimum, you want a cruise that goes to Glacier Bay. In 2007, the Coral Princess is sailing a great itinerary with long days in port, Glacier Bay and College Fjord. The prices aren't bad, either.
Princess and Holland also get the best spots in port. For example, while onboard the HAL Ryndam, we sailed past ultra luxury Seven Seas Mariner and docked in the best spot in Juneau. Meanwhile, the Mariner passengers were tendering into town.
Even if you don't choose Princess and Holland, it a cruise. How bad can it be?