Why look at power spectral density for stochastic processes? I have been told that for deterministic signals, it makes sense to look at their respective Fourier transforms/spectra. 

For stochastic processes on the other hand, I am supposed to work with power spectral density in terms of qualitative analysis. 

Why?
Because a stochastic process itself doesn't have a Fourier transform.

That's really all there is to it.

You can only transform signals (i.e. functions over a body isomorphic to $\mathbb R$, for example, functions of time). You can't transform a random variable whose individual realizations are such functions!