Body interference in my analog multiplexer, maybe AC electromagnetism induction?
the [Arduino pins wired to S0-S3 on the 74HC4067 pins are connected] but I'm not digitalWriting anything to them


That means that those Arduino pins are still in their power-up default state of inputs - as are the S0-S3 pins on the 74HC4067. This means you have inputs connected to inputs, without anything actively driving them into one logic state or the other.

On CMOS ICs, their inputs are typically such high impedance, that they can be influenced by external electric fields (e.g. from your body/hand), unless actively driven by an appropriate output or a pull-up/down resistor etc.  The behaviour and connections you have described fit with this expected result.

Assuming your schematic intends the LEDs to be off when S0-S3 = 0, then connect a pull-down resistor between each Sx pin and 0 V i.e. 4 resistors per 74HC4067 (10k is a reasonable resistor value for this purpose) and re-test.

Your 74HC4067 outputs should then no longer be affected by moving your hand nearby, even when the 74HC4067 inputs are not being actively driven by the Arduino, as those pull-down resistors will keep S0-S3 at logic 0 (unless they are actively driven to logic 1 by the Arduino, when you have programmed it to do so).