You can of course open the PSU, but you must observe some precautions. First (obviously), unplug the unit. Leave it unplugged for 15 minutes or so. In most PSUs there are bleeder resistors that will bleed down the stored voltages inside over time BUT THERE ARE NO GUARANTEES!!
When you open the unit, understand that there are two hazards:
- Stored high voltage (200V or so) on the "input" side of the unit.
- Stored high currents (tens of amps) on the "output" side of the unit.
The high voltage represents an electrocution hazard. The high current represents a burn hazard should something metal (such as your ring or watchband) touch the wrong terminals.
If you just need to move/secure a loose wire, you should be able to do that without touching any bare wires or terminals inside the box. If necessary use an insulating stick (eg, a tongue depressor or chopstick) to probe and move things around. Be wary of using something like a pencil (the lead is conductive) or a ball-point pen (which has a metal tip and often other metal parts). And, of course, screwdrivers are almost always metal and a hazard inside.
Be sure to close the unit back up before you apply power again. It's too easy to "jump ahead" and apply power with the unit open, then "remember something" and reach inside the live unit without thinking.