Any user can create a new folders within Dash, if they have the right permissions.
A user must have Contributor access to a folder in order to create a child folder within it.
If you would like a user to be able to create new top-level folders, that user must be assigned a permission group that gives "all folder" contributor access.
See this article to learn more about creating permission groups.
How can a contributor create a new folder?
Open up any asset, or select multiple assets and click the bulk edit button, so you are on the asset details page.
From here, find the Folders section in the Attributes Sidebar
Expand the folders drop-down menu, and create folders as follows:
To create a child folder, hover the mouse over the parent folder and click the "+" symbol to the right hand side
To create a parent folder, enter the name in the text field at the bottom of the folder list