We don't allow any Staff users to delete incidents, change requests etc.. If this is what you have in mind then you can do that by unchecking "delete" for each object under the Staff profile (and permission set if you are using).
If you want some Staff members to be able to delete and other Staff members to not be able to delete I think I would still do the above on your Staff profile and permission set and then create a new permission set called something like "allow delete". Set up that permission set to allow delete for each object and then assign it accordingly.
Hope that helps.