DESKTOP
Ok from the beginning.
I take a picture I use a 16gig sd card.
Insert that card into the desktop through its multi-card reader.
I then store them in an external HD through usb so I don't take up tons of space each being stored in a certain file.
I then take the photos and import them into LR 5.
LAPTOP
From the beginning.
I take a picture with the camera.
Insert the sd card into sd card reader built into the laptop.
Through usb the photos are transferred to the external HD.
I then plug that HD into the desktop to find them messed up as well.
All card readers are built in.
Never hook the camera up to computers via usb cable. I have always used the cards in the card readers.
Try connecting the camera to the desktop since that is the one you are having issues with and transferring the photos that way NOT using the built in card reader.
It is possible the card reader that is built in is not transferring files correctly and thus corrupting them.