"Ils ne boivent plus d'alcool."
Translation:They don't drink alcohol anymore.
The way I see it is that the ne....plus qualifies the verb (boivent), not the noun (alcool), so the translation will be they no longer drink alcohol or they don't drink (/are not drinking) alcohol anymore.
If you wanted to say that they are just not drinking more alcohol for the time being (e.g. they have switched to water for the rest of the evening) you could say ils ne boivent pas plus d'alcool (they are not drinking more alcohol) where the plus refers to the alcohol, not the drinking, and the ne....pas qualifies the verb.
However, just to complicate things, you could probably also use the first sentence (ne ....plus ) in the second situation to say that they were no longer drinking alcohol even if the implied time frame was just for the evening.
In this case, the verb boivent tells you it is plural since the singular would have been boit, but in some other cases such parle/parlent you would not be able to tell whether the pronoun was singular or plural when listening. But, as a general guide, you should always look at the verb for possible clarification if the subject pronoun is ambiguous.