The space $X\times\beta\mathbb{N}$ is normal if and only if $X$ is normal and $\mathfrak{c}$-paracompact. This follows from results of Morita (Paracompactness in product spaces, MR132525), which generalizes Dowker's characterization of countable paracompactness.
First note that $X\times\beta\mathbb{N}$ is normal if and only if $X\times K$ is normal for every separable compact Hausdorff space. This is because every separable compact Hausdorff space is a proper image of $\beta\mathbb{N}$.
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 of weight at most $\mathfrak{c}$. Hence, $X\times K$ is normal for every separable compact Hausdorff space 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 $X\times[0,1]^{\mathfrak{c}}$ is normal. Since the space $[0,1]^{\mathfrak{c}}$ is a separable compact Hausdorff space, this closes the implication loop.