You can set up this permission in the Admin -> Groups-> Folders area.
For each folder, you can choose the downloading rights for users in that particular group. If you would like users to have to request permission to download assets, then fill in the '2' permission value for each of the appropriate folders (if you have no additional folders, then just 'Universal').
This will mean that for each asset in that particular folder, the user will not see a download button in the asset details, and instead will see instructions for requesting approval.
Admin users will then receive an email indicating that a user has requested approval for downloading an asset, and will give you appropriate instructions.