Oil-dependent Alaska has long sought ways to fatten its coffers and move away from the fiscal whiplash of oil’s boom-and-bust cycles.

The newest idea, promoted by Republican Gov. Mike Dunleavy, would have the state capitalize on its oil and gas expertise to tap into a developing industry — carbon storage — as a way to generate new revenues without curtailing the industries that underpin Alaska’s economy. It’s also being pitched as a potential way for petroleum and mining companies to head off legal challenges over greenhouse gas impacts.

Hearings with state lawmakers are underway...