It’s a simple idea: To reduce carbon-dioxide emissions, put a price on them. But it’s dividing even those who agree that government should be pursuing policies to promote clean energy.

Supporters say making industries that burn fossil fuels pay to spit out carbon dioxide would encourage a shift to cleaner, alternative-energy sources by making it more expensive to burn coal, gasoline or natural gas. Some favor a cap-and-trade system, which sets a limit (the cap) on overall emissions and creates a market where companies can...