Here are a few possibilities:

Armée du Salut, Caritas, Centre social protestant, Emaus, La croix rouge, Women’s Shelter (Les Grottes).

Best wishes, R.

You can also put them in the big boxes labeled "Boîte à fringues" near most recycling zones. 

The used clothes will be given to the Coordination des Oeuvres d’Entraide Genevoises which forwards them to the charity organizations listed by Ritchie.

I sort my donations, giving clothes that are wearable and still reasonably fashionable directly to one of the neighborhood charity shops – I live in Eaux-Vives so there’s plenty of choice.  This means that the good stuff doesn’t get mixed up with the gear that’s only fit for “industrial waste” and saves the time of the voluntary workers who sort and sift donations.

And when I do have  “clobber” to discard, I use the “boîte à fringues”.

Something else that some charity shops are short of is clean Migros/Coop/Denner paper bags but not all, so ask first before taking them a bundle.