To use the Developer Console at all, you need the View All Data permission. 而 ** VIew All Data** 的权限可以在Profile当中去设置。

To run queries, you need to have the API Enabled permission.

To modify classes and triggers or run anonymous code, you need to have Modify All Data, Author Apex, and View Setup and Configuration.

To change Visualforce pages, components, Lightning, etc, you need the Customize Application permission.