sampling rate for band pass signal
Sampling is frequency domain convolution by a frequency domain impulse train with the impulses spaced apart by the sampling frequency. If there is overlap between the images thus created, then there will be information loss. As you can see in the illustration below, all of the sampling frequencies $f_s$ = 12 kHz, $f_s$ = 6 kHz, and $f_s$ = 4 kHz will work fine. I tilted the band tops to indicate positive and negative frequencies. The first curve is the signal's spectrum before sampling. The unit is kHz.



Using a small sampling frequency has the benefit that any processing will have less samples that need to be computed per time unit. For example filters designed to a performance specification can be lower-order than with a higher sampling frequency.

Sometimes it is useful if the signal is oversampled though. Things like polynomial interpolation work better because of the more dense sampling. A Hilbert transformer would have more room for the transition band. If processing is non-linear, it may be possible to avoid aliasing of harmonics by filtering after the non-linear processing; with critical sampling the harmonics might alias and contaminate the signal.

Using a low sampling frequency (6 kHz or 4 kHz) that aliases the bands may require more work later if you need to have the band back at the original frequency.

So the optimal sampling frequency depends on what you are going to do with the signal.