gn.general topology - Is there a "natural" characterization of when X × βN is normal?

The space $Xtimesbetamathbb{N}$ is normal if and only if $X$ is normal and $mathfrak{c}$-paracompact. This follows from results of Morita (Paracompactness and product spaces, MR132525), where he generalizes Dowker's characterization of countable paracompactness.

First note that $Xtimesbetamathbb{N}$ is normal if and only if $Xtimes K$ is normal for every separable compact Hausdorff space $K$. This is because every separable compact Hausdorff space is a perfect image of $betamathbb{N}$.

Morita's Theorem 2.2 shows that if $X$ is normal and $mathfrak{c}$-paracompact, then $X times K$ is normal for every compact Hausdorff space $K$ of weight at most $mathfrak{c}$. Hence, $Xtimes K$ is normal for every separable compact Hausdorff space $K$ since these all have weight at most $mathfrak{c}$.

Morita's Theorem 2.4 shows that a space $X$ is normal and $mathfrak{c}$-paracompact if (and only if) $Xtimes[0,1]^{mathfrak{c}}$ is normal. Since the space $[0,1]^{mathfrak{c}}$ is a separable compact Hausdorff space, this closes the implication loop.