"Onun düşmanı yok."
Translation:She does not have enemies.
It is optional here. The plural meaning is fully extracted from context :)
Nearly all English speakers form the negative of have (when it is the main verb, meaning "possess, own") with do as with most verbs, or they use the negative of have got instead.
So She does not have enemies and She has not got enemies / She hasn't got enemies would be the most common.
Directly negating have is rare - it sounds upper-class old-fashioned British to me. It's often not accepted here.
And She has not enemies or She hasn't enemies sounds wrong to me without any: She hasn't any enemies. But again, that construction is rare and I would advise to learn either She doesn't have (any) enemies or She hasn't got (any) enemies.
One possibility would be She has no enemies -- still a bit rare, but here no is used as a sort of combination of not any, i.e. negative + indefinite article.
Other than in a few conservative dialects of British English, this is not used in English. If you are using the verb "to have" as a main verb (not as a helper verb), you have to use "to do" in negation. In terms of "to have," we use the Oxford and Stand American rules.