(info from French consulate): If you work in France with your family, such an allowance is paid after the birth of a second child and until the child is 20 years old: http://www.ambafrance-uk.org/Social-Security

Mainly for families with two children or more: http://www.caf.fr

State unemployment benefits:

postdocs: this link provides some information in general terms about some of the state support available following the end of a work contract in France, though is focused here on CDI (i.e. permanent), rather than temporary (CDD) contracts, that form the basis of most postdocs. However, unemployment payments are available following the termination of a CDD as well, providing the period employment has been for at least four months: http://www.thelocal.fr/20130724/top-tips-if-you-end-up-jobless-in-france