Set up a group to which you add only the users for whom you wish to permit USB access, and restrict such access to members of that group.
Look at man sudo ans man sudoers for pointers on how to set up privileged commands (like mount) so only selected groups can use them.
Note that your distribution may be using udev rules to automatically mount USB devices, or that the automatic mount is being handled by, for example, the desktop manager. If that's the case you may need a different method to implement the restriction.
You might find it helpful to look at fora specific to your distribution for discussions of security issues. Or to ask the moderator(s) to move this thread to the security sub-forum on LQ.
Mentioning your specific distribution might also be helpful . . .