The facility has not operated since a 16-week gas leak began last October and was plugged in February. Residents are concerned by Southern California Gas Co.'s moves toward reopening it, citing health risks and potentially dangerous wells.
About 15 residents visited the state Capitol Wednesday to hold a rally and meet with Brown's staff.
Environmental groups say a shutdown of Aliso Canyon is good a chance to move toward renewable energy.
The facility is the largest natural gas storage plant in the state. SoCalGas has warned its closure could lead to blackouts.
Thousands of residents left their homes after the gas well blowout with headaches, nausea and nosebleeds.