To a graph $G$ one associates the binomial edge ideal $J_G$ generated by a collection of binomials corresponding to the edges of $G$. In this paper, we study the asymptotic behavior of symbolic powers of $J_G$, its lexicographic initial ideal $\mathrm{in}_<(J_G)$, and its multigraded generic initial ideal $\mathrm{gin}(J_G)$. We focus on the Waldschmidt constant, $\widehatα$, and asymptotic regularity, $\widehat{\mathrm{reg}}$, which capture linear growth of minimal generator degrees and Castelnuovo--Mumford regularity. We explicitly compute $\widehatα(J_G)$ and $\widehatα(\mathrm{in}_<(J_G))$, and compare the Betti numbers of the symbolic powers of $J_G$ and $J_H$, where $H$ is a subgraph of $G$. To analyze $\mathrm{in}_<(J_G)$ and $\mathrm{gin}(J_G)$, we use the symbolic polyhedron, a convex polyhedron that encodes the elements of the symbolic powers of a monomial ideal. We determine its vertices via $G$'s induced connected subgraphs and show that $\widehatα(\mathrm{gin}(J_G))=\widehatα(I_G)$, where $I_G$ is the edge ideal of $G$. This yields an alternate proof of known bounds for $\widehatα(I_G)$ in terms of $G$'s clique number and chromatic number.