I remembered another reason why closed surfaces of negative Euler characteristic cannot collapse under a lower bound on sectional curvature. 

Much more is true: if a sequence of $n$-dimensional closed manifolds $M_i$ of Ricci curvature $\ge -k^2$ Gromov-Hausdorff converges to a compact space of (Hausdorff) dimension $<n$, then the simplical volume of $M_i$ is zero for large $i$. 

Indeed, on the bottom of p.244 of of Gromov's <a href="http://www.ihes.fr/~gromov/PDF/7[35].pdf">Volume and bounded cohomology</a> he shows that a lower Ricci curvature bound implies that the simplicial volume is bounded above in terms of volume and the dimension. If a sequence of manifolds converges to a space of dimension $<n$ under the lower Ricci bound, then
their volume goes to zero; this is due to Colding, as mentioned e.g. on p.91 of <a href="http://library.msri.org/books/Book30/files/colding.pdf">
Aspects of Ricci Curvature.</a>

The simplicial volume of a closed surface of negative Euler charactersistic $\chi$ is $2|\chi|$, see p.217 in Gromov's paper, so it cannot collapse under a lower curvature bound. There are of course many high-dimensional manifolds of nonzero simplicial volume.