The folder properties item you are looking at does NOT mean the folder is read-only. It's just a control that can be used to set (or unset) files within the folder as read-only. Even though it's a checkbox it provides no status whatsoever. It's a legacy thing that you can safely ignore. Actual read-only folder settings are done through the Security tab of the folder, but you should not have to touch that unless you (or someone else) previously intentionally changed permissions using that tab.
Presumably you had an issue that led you to look at the folder properties in the first place. What problem occurred? What error message was displayed? What software were you using? Microsoft Office perhaps?