20-Year Prospect
The need to keep historical records will continue as long as history is made, and there will always be the need for people to record that history (for the #TBTs of the future, of course). Since many stenography jobs—like court reporters—are government funded, there will always be work as long as there are always taxes to pay for it.
Stenography will grow and change with technology. We've come a long way since pen and paper. Now, stenographers use machines with LED displays, WiFi connections, and on-board phonetic translators. As you can see, the field is doing just fine in the digital age.
But what about robots taking human jobs, you say? That's a great question. Now ask yourself, how reliable is that little lady inside your iPhone when you talk to her? Until she stops confusing "find pizza" with "I've got leeches," human stenographers probably don't have much to worry about. There are more important things for robots to do anyway.