The generic explanation is "...will be born with the genetic knowledge of the Goa’uld."
What does that mean and do they not already have access to that knowledge ? What more would the harsesis know, compared to a Goa’uld ?
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A goa'uld is a parasitic creature that lives inside a host, for example, a human. After that, the combined being is often called a Goa'uld, but they are still, in essence, two different creatures sharing one body.
The reason it is illegal to make an Harsesis is because you're effectively giving birth to a human (in the case of human-host Goa'ulds) with the knowledge of a Goa'uld, knowledge that could be used against the Goa'uld.
The harsesis is the result of the coupling of the Goa'uld hosts, and not simply the result of the usual reproductive process of the parasite. However, there would appear to be some genetic transfer from the host into the human child (in another episode we see that a coupling between Daniel and Hathor^ introduces human genetic material into the parasites making them compatible with human Jafar - so genetic transfer seems trivial).
So, the upshot is, you have a human which is also Goa'uld. This seems to be considered an abomination among the Goa'uld who prefer to simply take a host.
^ parasite reproduction is revisited later on when the Tok'ra queen is discovered. Unlike Hathor, this queen reproduces without a human host - but the offspring are purposefully made inert by not transferring genetic knowledge. This either shows that Goa'uld reproduction is quite complex, or that the showrunners decided that a lot of season 1 backstory needed a rewrite - including the Hathor mother of Goa'uld schtick