When a woman is pregnant with twins, how is it determined which sex is more genetically responsible?

Do you mean if the twinning was maternal or paternal? If that is the case then it is the general consensus that fraternal twins (non-identical) come from the mother's side (ie maternal, as two egss are fertilised at the same time) whereas monochoroinic twins (identical, where one fertilised egg splits into two) just happens and no one (except God) can take credit for them.
Hope that answers your question.

Fraternal twins are the twins that run in families. If you have boy/girl twins, the girl will most likely have twins herself, but the boy cannot pass this on to his wife/girlfriend but can pass it on to his daughter who will then most likely conceive twins. This is how twins skip generations.

Fraternal/Paternal are conceived when the women ovulates (produces eggs) twice in one cycle.

Identicals come from one egg that is why they are only a special gift from god to lucky parents.:)