In short, a pull down (or pull up) resistor with a low value makes the circuit low impedance, and the electrical noise will have less influence.
When a button is pressed, that connects the digital input pin of the Arduino board directly to 5V (or GND). Then it is almost impossible that electrical noise can get to that pin.
When a button is not pressed, then the resistor keeps the input pin low (or high). Then electrical noise can cause spikes.
A circuit with low impedance is less influenced by the electrical noise. The lower the value of the resistor, the less troubles with noise.
It is not something that can be calculated, see it as electrical/capacitive/electromagnetic/radio-noise combined with bad ground currents from a wrong ground path. What is the best way to receive that? With a high impedance circuit of course.
A pull down (or pull up) resistor with a value of 10k can be used when there are no wires connected to it that are going outside the circuit board. When there is a button with wires, then I would suggest a lower value, for example 2k2 or 4k7. Since you have troubles with noise, I suggest a value of 470 Ω or 1k.