Is it accurate to say in digital audio that, when a fader is down, then its value is "-$\infty$"?
If by "its" you mean the value of the fader, then yes - it's absolutely correct.
Fader defines the attenuation of the signal with respect to the reference level. The units are in logarithmic scale.

For example, no amplification (or in other words, multiplication of the signal by 1) corresponds to $0\; \mathrm{dB}$ since $10\log_{10} 1 = 0$.

If you want to completely attenuate your signal, then you have to multiply everything by $0$, hence $10\log_{10} 0 = -\infty$.