Is "Find a Steiner Tree of cost < K" NP-hard?
In fact, the optimization version of minimum Steiner tree isn't NP-hard, since only decision problems can be NP-hard. When we say that a minimization problem like minimum Steiner tree is NP-hard, what we mean is that the corresponding decision version is NP-hard. The decision version is just what you wrote: given an instance of minimum Steiner tree and a number $K$, to decide whether the minimum Steiner tree has cost at most $K$.