Translation:I do not want you to smoke in my apartment.
This sentence can be used - people actually say it if asked "What do you want from me?" ("Чего ты от меня хочешь?"). The translation offered by DL is likely to be used by a host explaining the rules to his guest/tenant.
this is much less likely to be said. would you say 'I want you to not smoke here?'
"I want you to not smoke here", whilst maybe more uncommon, seems perfectly fine to say.
In standard English, 'want' doesn't collocate with with "that"-clauses, it can only be followed by the so-called infinitive complex object (I don't want you to smoke)
I answered 'I don't want that he smoke in my apartment' which is correct as far as I can tell, and uses the subjunctive, so more literal, but it says it's wrong.