Technically one of the ketone groups exists as an enol due to chelation, but that's beside the point. You want to substitute the -OH group in the two aryl systems present in curcumin with -SH. Speaking from a purely theoretical point of view, doing by nucleophilic substitution would be exceedingly difficult as S
NAr requires deactivating groups. You could do by the benzyne mechanism, but that requires drastic conditions, so I don't know whether it would affect the other parts of curcumin. I'm no qualified professional.
What you could also try is, remove -OH by zinc dust and try to incorporate -SH group by electrophilic aromatic substitution. As OCH
3 is an activator in EAS, the electrophile will attack the ortho position - the position formerly taken by OH - and you will have the thiol. For more info on how to introduce a thiol group that way,
http://pubs.acs.org/doi/abs/10.1021/jo034013hThese are just my 2 cents. I repeat, I'm no qualified professional.