The Krull dimension of $C(X)$ is infinite for any infinite compact Hausdorff space $X$. By my answer here, it suffices to show there exists a continuous function $f:X\to\mathbb{R}$ which is not locally constant. To show this, let $A=\{x_n\}$ be a countably infinite discrete subset of $X$. For each $n$, by Urysohn's lemma there exists a continuous function $f_n:X\to [0,1/2^n]$ such that $f_n(x_m)=0$ for $m\geq n$ and $f_n(x_m)=1/2^n$ for $m<n$. The sum $\sum f_n$ then converges uniformly on $X$ to a continuous function $f:X\to[0,2]$, and $f$ is injective when restricted to $A$. Now let $y\in X$ be any accumulation point of $A$. Since $f$ is injective on $A$ and every neighborhood of $y$ contains infinitely many points of $A$, $f$ is not constant in any neighborhood of $y$.