"У мене немає батька."
Translation:I do not have a father.
Exactly for this reason you wouldn't use "the". It is only used in English to specify a particular thing when there could be many of them. In this case, a person can only have one father so there's no need to specify that it's THE particular father.
You wouldn't even use "the" with brother, sister, etc. because it doesn't make sense in that context. Having fathers, brothers, sisters is a general question, you either have them or you don't. You wouldn't ask someone if they had a specific brother or sister etc.
You can use it if you're talking about specific objects. Like if you asked someone to bring you a specific book, when they arrive you can ask them "Do you have THE book?".