by Richard Oelmann.
Well, you could try adding
.submissionsummarytable .ygtvtable {display:none;}(Note: NOT tested - should be tried out on a test environment first!)
But that will stop them seeing their submission files at all not just downloading them. Beyond that you are into editing core code to prevent the file area creating the submission as a link and you'll need someone else to advise on that.
However, I would also add to the previous comments that if this plagiarism is such a massive problem for your institution, then there would appear to be significant issues that need to be addressed that go way beyond blocking students downloading their own submission!