I think the option you are looking for is:
Setup -> Tickets -> Options -> Enable Cancellation By Client
Enable Cancellation By Client
Indicates under which conditions a "Cancel" option is available to the Client in the web interface and e-mail Ticket display.
Nope that wasn't it here is an example of what I am talking about
Where it has the delete ticket link I want that one gone the note ones to stay.
4/21/17 11:17 am
4/21/17 9:05 am
So, that looks like a dialog for a Tech user, not a Client user (because it has the option to add hidden notes and change the status of the tickets as well, which a Client can't do).
You could restrict Techs from having the permission to Delete Tickets. To do that you'd need to modify the default permission set (or create a new permission set and assign it to certain Techs) under Setup -> Techs -> Tech Permissions.
Full disclosure: I haven't yet tested this to see if changing that setting removes that option from the dialog you showed, but i'd suspect that it would (or would at least make it non-functional).