Is differential topology a dying field?

I don't think differential topology is a dying field.

I'll interpret this as the classification of smooth manifolds and, more broadly, maps between them (immersions, embeddings, diffeomorphism groups). Also, I'll restrict to the finite-dimensional case.

There are related topics which are very active, usually studying smooth manifolds with extra structure, e.g. exterior differential systems, foliations and contact structures, symplectic and Riemannian geometry. I won't comment much on these areas.

The classification of smooth manifolds was quite successful in the 60s with the h- and s-cobordism theorems framing many classification problems in terms of surgery problems. The classification of exotic spheres was more-or-less reduced to problems in homotopy theory, the stable homotopy groups and Kervaire invariant problems. The study of these invariants is still active, but the techniques are more algebraic. Moreover, there is still an industry of studying Riemannian metrics on exotic spheres.

Maybe one of the biggest open problems now in differential topology is the cobordism hypothesis, originally formulated by Baez-Dolan, but reformulated by Lurie. This is formulated as a classification of "fully extended topological field theories" in terms of $(\infty,n)$-categories. His sketch of proof is regarded as incomplete, and a few groups are trying to fill in the details. From discussions I've had with experts, a big issue here is foundational results in differential topology. Lurie's outline relies on results about manifolds with corners, and I think that Schommer-Pries has filled in some details, but I think that the proof of the cobordism hypothesis is still incomplete.

Another (very special) problem that has received some attention is the Hirzebruch Prize Question:

Does there exist a 24-dimensional compact, orientable, differentiable manifold $X$ (admitting the action of the Monster group) with $p_1(X) = 0$, $w_2(X) = 0, \hat{A}(X) = 1$, and $\hat{A}(X, T_C) = 0$?

Here $\hat{A}$ is the A-hat genus. The twisted Witten genus is supposed to be related to certain modular functions (McKay-Thompson series) associated with Monstrous Moonshine. I believe that Hopkins proved that a manifold with the right properties exists, but only in the topological category, and without the action of the Monster group. Daniel Allcock is working on constructing this manifold.

Shmuel Weinberger has championed the study of decidability questions in differential topology.

The Novikov conjectures would imply that $\mathcal{L}$-classes (certain combinations of Pontryagin classes) are invariant under homotopy equivalence of smooth aspherical manifolds. See a recent survey.

There is still active study of diffeomorphism groups. An active topic here is the study of homological stability for diffeomorphism groups, which is an understanding of the homology of the classifying spaces for such groups.

Probably it is the blanket term "differential topology" which is dying, as people use more specific terms to describe different aspects of the study of smooth manifolds and maps.