Addiction is most often a problem of self-control. If one is not capable of factoring in the future consequences of one's actions, one way to make this is taken into account is to distort prices appropriately. This is what taxes (and subsidies) are good at. While we know rather well how to design taxes on externalities born by others or the community, the case is more difficult for externalities inflicted on future selves.
Luca Bossi, Paul Calcott and Vladimir Petkov get on the case i the context of externalities, self-control issues and imperfect competition, as applicable for cigarettes and their highly concentrated industry. They also implement time-consistent taxes to accommodate the addiction, which means that people or government would not want to deviate from the social optimum. Taxes are thus state dependent and described by a rule. One important result is that combining addiction and imperfect competition leads to lower taxes that previously reported, because prices are already higher to start with if providers are oligopolistic. Were some drugs to be legalized, one has thus to keep in mind that the new market structure matters in the design of the new taxes.