There are many different ways governments can address negative externalities, and there is really no absolute one-size-fits-all policy for every imaginable circumstance.
One often-touted solution is market pricing of negative externalities, an idea often raised in the context of carbon pricing. Many people advocate this, and it has the advantage of fairly mediating between pricing the carbon created by the developed world and allowing trading to benefit developing countries that are harmed by climate change and produce less per capita carbon. However, accurate carbon pricing, obtaining global cooperation, and assessing effectiveness often run into practical roadblocks.
Regulation, such as increasing gas mileage and bettering pollution standards for cars, when applied gradually and sensibly, is a fair and measurable way to reduce harm, if handled by expert economists and scientists rather than by political grandstanding. Forbidding companies to poison the air we breathe and water we drink falls under the basic remit of a government to prevent entities from actively harming citizens. Regulation against poisoning an aquifer is a logical extension of criminal laws that make it illegal to drop poison in someone's drink over dinner. A "negative externality" of lead in drinking water that kills or sickens children, as is happening in Flint, Michigan, is just as harmful as a murderer in a crime film trying to kill a victim with arsenic.
Taxing consumers can "nudge" people to better behavior, and tax regimes often are used as ways of reducing individual behaviors with negative externalities. One successful version of this is the use of congestion pricing in city centers to handle excessive vehicular traffic. The main concern with this method is that it places an undue burden on the poor while allowing the wealthy to continue to impose negative externalities on their environment.