We consider node-weighted network design in planar and minor-closed families of graphs. In particular we focus on the edge-connectivity survivable network design problem (EC-SNDP). The input consists of a node-weighted undirected graph G = (V,E) and integral connectivity requirements r(uv) for each pair of nodes uv. The goal is to find a minimum node-weighted subgraph H of G such that, for each pair uv, H contains r(uv) edge-disjoint paths between u and v. Our main result is an O(k)-approximation algorithm for EC-SNDP where k = maxuv r(uv) is the maximum requirement. This improves the O(k logn)-approximation known for node-weighted EC-SNDP in general graphs . Our algorithm and analysis applies to the more general problem of covering a proper function with maximum requirement k. Our result is inspired by, and generalizes, the work of Demaine, Hajiaghayi and Klein  who gave constant factor approximation algorithms for node-weighted Steiner tree and Steiner forest problems (and more generally covering 0-1 proper functions) in planar and minor-closed families of graphs.