I am an EU citizen (not French), but my spouse is not. We are considering moving to France, where initially only I would have a job.
What choices would my spouse have for health insurance and what would be the associated costs?
If you are yourself insured through one of the statutory health insurance organizations (which is almost certainly the case if you are employed in France), your spouse and children can also be covered at no extra cost, no matter their citizenship (this is called being an “ayant-droit” or beneficiary). Some paperwork is however required.
If your spouse finds a job, he or she would then be covered by the relevant health insurance organization and would have to pay the mandatory contributions (for employees, those contributions are automatically withheld from wages).
Additionally, if someone cannot be covered through any other means but resides legally in France for more than three months, they are usually entitled to free health insurance through the couverture maladie universelle (but there are some exceptions like diplomats). See also service-public.fr. It's not really relevant here but there is also a distinct program to provide medical insurance to people who are staying illegally.
On the other hand, if you aren't insured through the French statutory insurance system yourself (say you are working for an international organization, you are a posted worker, etc.), I am not sure if your spouse can still benefit from it.