I just took a normal (fairly light) water-resistant coat, along the lines of a windbreaker, and an umbrella when I went. It was generally quite sufficient. It's not as though getting wet is the end of the world; you can always change into dry clothes when you get back.
Alaska is a big place, and the weather conditions vary a great deal across it. I think the most rain I encountered was actually (and very much atypically) around Deadhorse and along the Dalton Highway.
Late summer and early fall can start to get rather chilly in Alaska and the Yukon. A sweater and/or sweatshirt would be a good idea. I had to buy a sweater in Whitehorse, but fortunately found a nice comfortable one for a good price, so it turned out very well. In fact, it's now one of my favorite sweaters.