Only the owner of an Organization can adjust the user rights. To adjust a user's role, go to "Organizations" on the top left corner of your screen and select the organization you want to adjust.
Next select the "View Members" button.
A list of all members of this organization will be displayed. Here you can select the user you want to adjust roles for. When selected, the role of this user can be chosen. Default or Read Only.
With Read Only, he/she will not be able to upload files to this organization.