The 74HC4051 and its variants such as the CD74HC4051 have an input called INH or ~E depending on the manufacturer.
Typically you should hard-connect this pin to GND in order to always enable the multiplexer.
Whenever you want to disable the multiplexer (i.e. none of the inputs is connected to the output, connect this pin to Vcc temporarily.
Source: This NXP datasheet, section 7.