I believe the result holds quite generally. (In particular, you don't need any assumption about the projective dimension.)

Any ring spectrum $\mathbb{E}$ induces a homological functor

$ \mathbb{E}_*(-) : SH^{fin} \rightarrow \mathbb{E}_\ast\text{-grMod}$

from the stable homotopy category of finite spectra to the category of $\mathbb{E}_*$-graded modules (
where $\mathbb{E}_\ast$ is the coefficient ring of $\mathbb{E}$).

I believe that the collection of those finite spectra $X$ such that $\mathbb{E}_*(X)$ is coherent as a graded $E_\ast$-module can be shown to be a thick triangulated subcategory of the stable homotopy category of finite spectra using properties of coherent modules and the fact that $\mathbb{E}_*(-)$ is a homological functor.

If $\mathbb E_{\ast}$ is a coherent ring then this thick triangulated subcategory contains the sphere spectrum. But since the thick triangulated subcategory generated by the sphere spectrum is the whole of $SH^{fin}$ it follows that $\mathbb E_\ast(X)$ is a coherent $\mathbb E_*$-module for every finite spectrum $X$.

In other words, if $\mathbb{E}$ is a ring spectrum whose coefficient ring is coherent then $\mathbb E_\ast(X)$ is a coherent graded $\mathbb{E}_\*$-module for any finite spectrum $X$.

Here's a paper that might be of interest:
 
 o Pierre Conner & Larry Smith - On the complex bordism of finite complexes (1969)