There is a nice characterization of the image of the norm map of $F$ over $K$, where $F$ is an unramified extension over $K$. More precisely, $N_{F/K}(F^{\times})=\lbrace u \pi^n| u\in \mathcal{O}_K^{\times} , n\in f\mathbb{Z} \rbrace$, where $\pi$ is a uniformizer of $K$ and $F$ and $f$ is the residue degree of $F/K$. The results can be found from Jacobson's Basic Algebra II p.610.
I'm wondering is there similiar characterization for the case of totally ramified extension?
There won't be a characterization depending only on the degree, because there can be more than one totally ramified extension of any given degree (while there is a unique unramified degree $d$ extension for every $d \ge 1$).
That said, the answer is perhaps even more beautiful than you suggest. Take any finite-index open subgroup $U$ of $K^\times$. Then there is a unique abelian extension $F / K$ such that $N_{F / K}(F^\times) = U$, and its Galois group is canonically isomorphic to $K^\times / U$. The extension $F$ will be unramified if $U$ contains $\mathcal{O}_K^\times$, and totally ramified if $U$ contains a uniformizer of $K$.
This correspondence between abelian extensions of $K$ and finite-index subgroups of $K^\times$ is one of the main results of the subject called local class field theory; there is a very nice exposition of this theory in the first few chapters of Milne's notes.