Cost-effective, fast, and reliable DNA sequencing can be enabled by advances in nanopore-based methods, such as the use of atomically thin graphene membranes. However, strong interaction of DNA bases with graphene leads to undesirable effects such as sticking of DNA strands to the membrane surface. While surface functionalization is one way to counter this problem, here, we present another solution based on a heterostructure nanopore system, consisting of a monolayer of graphene and hexagonal boron nitride (hBN) each. Molecular dynamics studies of DNA translocation through this heterostructure nanopore revealed a surprising and crucial influence of the heterostructure layer order in controlling the base specific signal variability. Specifically, the heterostructure with graphene on top of hBN had nearly 3-10× lower signal variability than the one with hBN on top of graphene. Simulations point to the role of differential underside sticking of DNA bases as a possible reason for the observed influence of the layer order. Our studies can guide the development of experimental systems to study and exploit DNA translocation through two-dimensional heterostructure nanopores for single molecule sequencing and sensing applications.