"They don't drink alcohol anymore."
Translation:Ils ne boivent plus d'alcool.
Could you explain what you mean with your sentence?
1) They don't drink alcohol anymore → They drank before. Now they don't. They won't drink ever again. → Ils ne boivent plus d'alcool.
This is the construction "ne... plus", meaning "not anymore" or "never again."
2) They don't drink any more alcohol → They stopped drinking for the moment. They may drink alcohol some other time or they drank something else besides alcohol. → Ils ne boivent pas plus d'alcool.
This is the negation "ne...pas", meaning "not", combined with an adverb of quantity "plus de", meaning "more". Together it means "not more".