What's the safest way to ship diskettes?
Any time I shipped magnetic media, I used the anti-static bags that electronic components came in.  Granted, when you have access to an IT department, they are easily available; we would save them for reuse.  I still do, but my stash isn’t as plentiful as it used to be.



However, you can get reusable zip poly anti static pouches.  They will provide shielding from low level EM fields and are transparent enough that if your package is opened for inspection, the contents are easily identifiable.  I’ve never had any issue with data loss using this method whether shipping in-country or internationally.

You didn’t mention which format the diskettes were; if they were 3.5”  or 5.25” or even the large IBM 8” but typically you could copy the disks so long as the drive was capable of writing it.  For instance, the 3.5” 1.44MB FDD drive could easily handle the 720K disks.  We had a “copyboard” in an IBM AT that would duplicate anything including copy protected disks for Apple.  It didn’t even matter if the OS could read it so long as the drive could.  

Along that note, I just recently ran across some Gateway 2000 MS-DOS 6.22 setup disks in a storage box that weren’t exactly put away with the greatest of care (about 25 years old).  I was able to image them and install MS-DOS onto a Virtual Box VM with no issue.  I’d bet your disks are still readable.  If at all possible, I’d make an image for safe keeping (I can walk you through it, but it’s technically a different question).  Oh, and those DOS disks were imaged using a Mac!