Sampling the Wave Equation for String Vibration

Assuming I understand the question correctly, you need to sample the string's position at least twice as fast as the maximum frequency you anticipate the string can move or the highest frequency you care about. So if the highest overtones the string produces are 16kHz, you need to measure at at least 32kHz if you want to be able to reconstruct the original motion faithfully. This is the Nyquist-Shannon sampling theorem and you can probably find several hundred descriptions of it here and elsewhere on the internet.

If you know the lowest frequency the string can produce or that you care about, there are other tricks you can do (look up heterodyne) to reduce your sample rate requirements assuming you have the equipment to do so on the analog side.