We consider a new Steiner tree problem, called vertex-cover-weighted Steiner tree problem. This problem defines the weight of a Steiner tree as the minimum weight of vertex covers in the tree, and seeks a minimum-weight Steiner tree in a given vertex-weighted undirected graph. Since it is included by the Steiner tree activation problem, the problem admits an O(log n)-approximation algorithm in general graphs with n vertices. This approximation factor is tight up to a constant because it is NP-hard to achieve an o(log n)-approximation for the vertex-cover-weighted Steiner tree problem on general graphs even if the given vertex weights are uniform and a spanning tree is required instead of a Steiner tree. In this paper, we present constant-factor approximation algorithms for the problem with unit disk graphs and with graphs excluding a fixed minor. For the latter graph class, our algorithm can be also applied for the Steiner tree activation problem.