Welcome to Alaska----
You didn't mention which part(s) of Alaska you are going to visit in June. Are you going to cross the gulf and spend some time in Central Alaska or will your land portion be in Southeast Alaska? Also, big difference between very early June and late June.
For early June you shouldn't have to pack any different than you would for a Baltic or a New England cruise that time of year. In late June many of the younger visitors in central Alaska are wearing outdoor shorts, it can hit over 90 F during the days then in Central Alaska.
Since you might get into a stretch of fairly warm weather at times dressing in layers is appropriate. The mosquitoes are pretty much not a problem in Southeast Alaska unless you get away from the coast and in Central Alaska, in early June, not a problem either, usually. However, in late June, watch out so don't forget the bug dope, treat the mosquitos with respect, as they are our state bird, LOL.
You might run into a bit of rain, particularly on a shore excursion in Southeast, so like us bring along fold-up light wind breaker rain jackets. Umbrellas more often than not are a joke because if you have rain you usually have wind that destroys them. I prefer some real light fold up rain pants, which you probably will never use but if you need them they really are nice. For shoes, I always bring my running shoes and if we are going to get into wet terrain or do some trail walking or climbing I bring my lighter waterproof hiking boots (note: hiking boots not hiking shoes).
A light sweater is great but even better is a hooded sweat shirt and a couple of baseball caps. Bring binoculars. Light all night pretty much so no need for flashlights. Sun glasses you might need, especially around the glaciers, but I wouldn't worry about sun screen unless you plan on glacier hikes or some mountain climbing above snow line. Here is a list a lady provided who had made 10 cruises to Alaska:
"2 pair jeans (in case one gets wet)
3-4 shirts to wear with jeans
one pair capris to wear onboard (sea days)
1 nice dress for formal nights
1 or 2 skirts and/or nice pants and shirts/sweaters to wear with them
1 hooded sweatshirt
1 0r 2 zip up sweatshirt-like jackets or sweaters
1 light waterproof jacket or warm vest… or poncho
scarf, hat and gloves
1 or 2 pair dress shoes/sandals
1 pair sandals for pool deck
swimsuit and coverup
2 pair good walking shoes
socks and underwear
small backpack (easier than carrying a purse)
sunscreen (glare off boat can be strong)
Of course, this is a female list…just make the necessary adjustments for males.
Many are probably thinking…seriously? a bathing suit? Yes! The ship’s pools will be heated and the hot tubs will be running….enjoy them!"