We consider wireless sensor networks under a heterogeneous random key predistribution scheme and an on-off channel model. The heterogeneous key predistribution scheme has recently been introduced by Yağan - as an extension to the Eschenauer and Gligor scheme - for the cases when the network consists of sensor nodes with varying level of resources and/or connectivity requirements, e.g., regular nodes vs. cluster heads. The network is modeled by the intersection of the inhomogeneous random key graph (induced by the heterogeneous scheme) with an Erdős-Rényi graph (induced by the on/off channel model). We present conditions (in the form of zero-one laws) on how to scale the parameters of the intersection model so that with high probability all of its nodes are connected to at least $k$ other nodes; i.e., the minimum node degree of the graph is no less than $k$. We also present numerical results to support our results in the finite-node regime. The numerical results suggest that the conditions that ensure $k$-connectivity coincide with those ensuring the minimum node degree being no less than $k$.