Related perhaps to the question asking broadly why a living being would be a poor Horcrux choice, but more specifically: what would happen if it reproduced? Would the resulting baby have a piece of the Horcrux creator's soul? Would it trickle across generations like a carried gene? Or does the curse remain solely with the initial intended target?
Nothing. A Horcrux is a piece of one's soul put into an object (or a living thing). When something reproduces, nothing happens to its soul, so I don't see how anything would be particular to its offspring, except for the fact they would be especially mean and annoyed, as a side effect of hanging around a horcrux all day. I don't think there is any canonical content to back this up, so if there is give me a heads up, but I'm coming from the definition of a horcrux: A piece of someone's soul which is broken apart from the whole and hidden inside something else (be it an inanimate object or a living being) in order to protect it from harm. By this definition, and the knowledge that a being doesn't "inherit" its parent's soul - since a soul isn't exactly a biological part of a being - we can say that nothing will happen to the horcrux or to its offspring.
The Horcrux itself would stay within the being it was sealed in.
A piece of soul, once bound to an object, is then dependent on that object and can’t live outside it or exist without that specific object which has become its enchanted body.
“But my point is that whatever happens to your body, your soul will survive, untouched,’ said Hermione. ‘But it’s the other way round with a Horcrux. The fragment of soul inside it depends on its container, its enchanted body, for survival. It can’t exist without it.”
- Harry Potter and the Deathly Hallows, Chapter 6 (The Ghoul in Pyjamas)
A Horcrux bound to a living thing would work in the same way, so the piece of soul would therefore have to remain in the original living being it was sealed in - it can’t transfer to its offspring. Though it’s possible being born of a Horcrux may affect the offspring in some way, it won’t be by transfer of the soul piece or part of the soul piece, because it’s bound to and dependent on the original being.