Which method of controlling speaker volume consumes more power?
I am assuming you mean that the speakers have built-in audio amplifiers.

The power consumed by the audio will depend on the volume of the loudspeakers. If both amplifiers are playing at the same volume they'll both consume the same power.







simulate this circuit &ndash; Schematic created using CircuitLab

The power dissipated in a 10 kΩ pot with a 1 V line-level signal would be given by \$ P = \frac {V^2}{R} = \frac {1*2}{10k} = 0.1~mW \$. Maybe you could save 0.25 mW by turning the pot up to max and turning the volume down in the computer. Compared with the several watts require for the amplifier it's negligible.