My husband is from Anchorage, Alaska so we head to Anchorage to visit family every summer. I've climbed glaciers in sneakers and shorts and a sweatshirt in the pouring rain (pop up rain - I certainly didn't dress that way knowing that it would rain)... it's never really "that cold" in Alaska during the summer so even when you're wet from rain you're not freezing.
When doing things outdoors in Alaska it's best to always have your hands free (just in case

) so I'd recommend rain coats with hoods and good climbing/walking shoes or sneakers with good tread. If all else fails put baggies inside of your shoes to keep your feet and from getting wet. (That's an old trick from growing up in Pennsylvania and playing outside is snowsuits and boots!) Above all, enjoy! Alaska is beautiful and majestic!