Why do HK chefs let water faucets run unceasingly, when the basin overflows?
This appears to be a dipper well: a continuously running sink used to rinse utensils. The water runs continuously to remove contaminants, always leaving a clean supply of water to rinse dirty utensils and, in this case, to clean out the wok.

They are rather wasteful: the linked article states they use 30-60 gallons (110 to 230 L) of water per hour.