Let $(M,g)$ and $(M',g')$ be non-orientable Riemannian surfaces with fixed boundary $\Gamma$ and fixed Euler characterictic $m$, and $\Lambda$ and $\Lambda'$ be their Dirichlet-to-Neumann maps, respectively. We prove that the closeness of $\Lambda'$ to $\Lambda$ in the operator norm implies the existence of of the near-conformal diffeomorphism $\beta$ between $(M,g)$ and $(M',g')$ which does not move the points of $\Gamma$. Hence we establish the continuity of the determination $\Lambda\mapsto [(M,g)]$, where $[(M,g)]$ is the conformal class of $(M,g)$ and the set of such conformal classes is endowed with the natural Teichm\"uller-type metric $d_T$. In both orientable and non-orientable case we provide quantitative estimates of $d_T([(M,g)],[(M',g')])$ via the operator norm of the difference $\Lambda'-\Lambda$. We also obtain generalizations of the results above to the case in which the Dirichlet-to-Neumann map is given only on a segment of the boundary.