There are two parts:

As Avinash points out, the default mode 'fan_in' is probably a good choice.
For some intuition of why this is: Each output is a weighted sum of fan_in inputs. For linear, this is one row of the matrix multiplication, for convolutions it is number of in-channels  * kernel size.

When @ptrblck and I implemented StyleGAN 16 for PyTorch (but I don’t think the StyleGAN authors necessarily invented it), I’ve come across the idea of not having the multiplier used during init, but applying them to the weight before using it. This has the effect of using the scaling for both initial values and gradient updates. (Even if it is not that efficient without a hand-made convolution kernel.)
Best regards
Thomas