Reasonable? Not really.
Moreover, if I'm not mistaken, being proper isn't preserved by continuous surjections; for example assuming $CH$ there is a continuous map from the space $\omega^\ast$ (which is proper) onto the space associated with the po-set which shoots a club through a stationary/co-stationary subset of $\omega_1$ (contrast this with the $c.c.c.$ which is preserved by continuous surjections.) This (at least in my opinion) makes the idea that there is some useful topological characterization of proper a moot point.
That said, the topological interpretation of forcing has had an influence on how proper forcings are cooked up. The relevant topological observation being that the only forcings worth playing with, come from spaces which are completely regular, which are comfortably embedded as subspaces of the product spaces $[0,1]^S$ (with $S$ some infinite set; normally taken to be an ordinal.) This is desirable, since in this case, the base for the product topology admits a convenient coding using elements of $\mathsf{Fn}(A, \cal I)$ (where $\cal I = \{ (a,b)\cap [0,1]: a< b\in \mathbb{Q}\}$.)
Anecdotally, I find that type of perspective/coding makes tradmark tools of the trade like side-condition style forcings (i.e. $\in$-chains of models with countable/finite working part), promises, creatures, souslin-c.c.c./proper, ... easier to digest.