Many museums offer free or reduced admission for students and anyone under 18, although sometimes you have to ask. Make sure to bring a student ID card.
There are hostels in most towns in Alaska. Except for those in Anchorage, Fairbanks, and Juneau, they tend to be open in the summer only. I have described hostels in the text for each town. Few are affiliated with organizations such as Hostelling International that provide discounts to members. For a comprehensive listing of hostels, go to www.hostels.com and navigate to Alaska by clicking on the maps.
Many students travel to Alaska for summer work. It's usually possible to get a job in a fish cannery in most coastal towns. Work on the slime line is hard and unpleasant and the pay is low, but if the season is good and you work long hours, camping to keep your expenses low (often canneries have tent cities of summer workers nearby), you can take home more money for your summer's work than you would normally earn elsewhere. Hillary Rodham Clinton worked an Alaska slime line when she was in school, and you can, too (George W. Bush worked in Alaska as a young person as well, at a politically connected air service and construction company). Offshore fish-processing ships hire many unskilled workers, too, but I wouldn't recommend that for a student; once you're on the ship, you are trapped there, and you may hate it. The work is hazardous (federal worker safety and minimum wage laws do not apply) and medical care may be inadequate. Also, don't come north expecting to make fabulous wages. The stories of college students making huge crew shares on fishing boats are legends -- there are experienced deckhands to take lucrative jobs before boats hire raw hands they have to train. Go that way for the adventure (and danger and hard work) but not out of greed. Jobs are always available in the tourism industry, too, although without the extreme flexibility of fish processing. You could be a tour guide or, more likely, a chamber maid or restaurant server.
The Alaska Department of Labor posts a lot of useful information on its website, including job applications for seafood work and a live database of openings; start at http://alexsys.labor.state.ak.us.