Learn English – would like + infinitive (negative form)

It's correct; it has a doubling of negatives, but one that all but the most disapproving of double negatives would allow.

It's logically equivalent to "I would like to have children", but differs in that it is reacting to the idea of not having children rather than to the idea of having them. While those amount to the same thing logically, ones emotional response to each idea could be of different degree or contradictory (it is possible for one to say they wouldn't like to have children, but also wouldn't like to not have them).

For this reason, the double-negative expresses something than the negative-free equivalent, and most would allow it.