What's the cheapest way to detect vibration with Arduino?
Piezo sensors are cheap, reliable and designed for the purpose you suggest - arduino tutorial

In the uk, bitsbox has one for 75p. You can probably find others cheaper, or salvage from electronic toys.

EDIT following poster's clarification of use:
Sound is vibration through the air, I see no reason why a piezo sensor cannot accomplish what you suggest, but the form factor may not be ideal for fixing to ping pong nets! I think a flex sensor would be more suitable, sewn into the top of the net.