The volume control can be implemented as a multichannel analog attenuator, after the crossover, or in the mathematics of DSP crossover itself.
In theory, analog volume controls are superior to digital volume controls. Analog controls reduce the noise level along with the signal level. Digital volume controls reduce the signal level but the noise floor remains the same (and then, only if the signal is properly dithered after attenuation).
In practice, as long as the noise floor of the DAC is below the noise floor of the listening room (or of the other electronics), it is a non-issue. And with modern DACs, this is fairly easy to achieve. In addition, digital volume controls are superior in terms of channel tracking and attenuation accuracy.