The more popular rivers, Thorne, Harris, Karta are an hour or more from Craig but the Klawock river is only about 8-10 miles from town and gets silvers, and a decent run of sockeye, haven't fished it but know guys that do okay on the Klawock.
Most of my fishing on the island has been on the Sarkar (oops, just gave up a zipper) for trout. And on the no-name stream that has the waterfall the resort was named for, good steelhead fishing there right now thru end of May, good cutthroat fishing above the falls, but don't tell anyone.