Stash doesn't seem to do much with submodules. From what I've seen, if a repository has a submodule, Stash simply displays the .gitmodules file. I think there are a few things Stash could do.
From a user perspective:
1. An option to show the submodule folder (perhaps in a different style, or annotated as a submodule in some way) and files based on the commit state of the main repository.
2. An option to list submodules for a project and link to the underlying repository
For example, github shows the sizzle submodule in https://github.com/jquery/jquery/tree/master/src with a link to the repository for the submodule and includes the current changeset.
From an administrative perspective (I suspect some/all of these may not even be possible based on my admittedly limited understanding of Git):
1. Configure repository permissions separately for when a repository is used as a submodule (e.g. read-only when used as a submodule, but full permissions when used as a regular repository).
2. An option to have Stash assume "--recursive" behavior for clone operations
3. An option to keep submodules in sync when a user performs a pull or rebase (the equivalent of forcing a submodule update)