How do I know if the serial cable is plugged in the right way?
The data sheet for the part labels it.



Of course, as far as I can see online, the front of the board does not have any silkscreen. But the right most pin has a square pad. This is typically pin 1. And even on the blurry picture, its plain to see that that pin has thermal reliefs connecting it to the ground plane. Easy bet that its ground, and the rest of the pins will match the data sheet now that you have identified it.