The Princess website has a pdf file for the port of Whittier that lists 'short term' luggage storage but I believe that is for people starting a cruise there and not for debarked passengers. The file doesn't say one way or the other. Many tour operators will store your luggage for you, or obviously take it along for the ride.
The train depot in Anchorage is about 15 minutes from the airport. Taxis and shuttles are available at the train depot. A taxi will be around $20.