Let $p \in X$ have codimension $n \geq 1$. There is a maximal chain of irreducible closed subsets $\overline{\{p\}} = C_n \subsetneq \dotsb \subsetneq C_1 \subsetneq C_0 = X$. Each $C_i$ has codimension $i$; in particular $C_1$ is an irreducible closed subset of codimension $1$. Let $q$ be the generic point of $C_1$, i.e., the unique point so that $C_1 = \overline{\{q\}}$ (see, for example, https://math.stackexchange.com/questions/1155342/irreducible-closed-subsets-of-a-scheme-corresponds-to-points). Then $q$ is a point of codimension $1$ and $p$ lies in the closure of $q$.

Let $p \in X$ have codimension $n \geq 1$ (edit: with $n$ finite). There is a maximal chain of irreducible closed subsets $\overline{\{p\}} = C_n \subsetneq \dotsb \subsetneq C_1 \subsetneq C_0 = X$. Each $C_i$ has codimension $i$; in particular $C_1$ is an irreducible closed subset of codimension $1$. Let $q$ be the generic point of $C_1$, i.e., the unique point so that $C_1 = \overline{\{q\}}$ (see, for example, https://math.stackexchange.com/questions/1155342/irreducible-closed-subsets-of-a-scheme-corresponds-to-points). Then $q$ is a point of codimension $1$ and $p$ lies in the closure of $q$.

(Edit: This can fail if $n=\infty$, as in R. van Dobben de Bruyn's answer.)