What is a good value for a pull-down resistor, which when switched, connects to 24v? Is there a formula to derive this?
This is effectively how it's drawn in the schematic (the inverter input's impedance is 650kohm):
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10K is often a good choice if you don't know what resistor to try.
Simulate or think about the circuit with a 10K resistor. are the currents that will flow apropriate to the devices and the power-supply? will the voltages be good? if not adjust it.
resistor choice for pull-ups and pull-downs is often not critical.