Which one would save more power charging a phone?

1.) Charging the iPhone with a USB hooked to a desktop that is in use.

OR

2.) Charging the iPhone from an outlet with the USB and box?

1. Assuming the desktop PC would have been in use anyway.

If you use the computer's USB ports to charge your phone, your computer takes more power.
I would charge your phone with a USB wall outlet adapter.

Without knowing the power transfer efficiencies of the two devices, it is difficult to say exactly. The AC adapter is such a cheap device that it could be less efficient than a whole computer for just charging a phone.

Both would be about the same, assuming efficient power supplies.

Both have internal power supplies that convert line AC voltage to the 5 volts needed for the USB output of 5 volts DC.

What would be lower efficiency would be charging the phone from a USB outlet on a portable PC that is running off it's battery.

Desktop. If already in use, the additional power sent to the phone wouldn't add much additional load on the system. The components in the wall adapter would waste some energy, and since the only purpose would be to charge the phone, all the wasted energy would be the 'fault' of the phone.
But if the computer wasn't being used for other purposes, it would be far more wasteful, and would take longer.