Let $R=k[x_1,...,x_{n},y_1,...,y_n]$ be a polynomial ring over a field $k$ and $I=\langle \{x_iy_j|$ for some $i,j \in\{1,...,n\}\}\rangle$ be ideal of $R$ and there are $r,s\in\{1,...,n\}$ such that $x_ry_s\notin I$. Assume $R/I$ is a Buchsbaum ring. Suppose $(R/I)_p$ is Cohen-Macaulay for all $p\neq \langle x_1,...,x_{n},y_1,...,y_n\rangle$. Can we say that $R/I$ satisfies the Serre's $(S_2)$ condition?
See here (Theorem 1.3, (c) implies (d)).
This answers the question for the case when $I$ were the edge ideal of a non-complete bipartite graph.
The only thing to prove is the following: $\operatorname{depth}R/I\ge\min(2,\dim R/I)$. It's obvious that $\dim R/I\ge 2$, so we have to prove $\operatorname{depth}R/I\ge2$. Since the graph $G$ is supposed non-complete the simplicial complex $\Delta_G$ is connected (see Lemma 1.2 from Bipartite $S_2$ graphs are Cohen-Macaulay). The result we use now says:
Let $Δ$ be a simplicial complex on the vertex set $V$ and $|V|\ge 2$. Then $\operatorname{depth}k[Δ] \ge 2$ if and only if the geometric realization of $Δ$ is connected.
For a proof see Hibi's paper Union and glueing of a family of Cohen-Macaulay partially ordered sets, Example A, page 8.