The web will be here for at least another 10 years, so that's certainly not an issue. I'll try to highlight some pros and cons.
The main risk is that the number of available roles decrease for other reasons; for example, web development becomes more productive because the base platforms and frameworks become richer. The other is that there is something of a startup bubble right now, which makes it a great time to find a good demanding intern type role, but which again doesn't guarantee continued employment.
The key is really... what do you love? If you're motivated, you'll accelerate in skills that will keep you in demand regardless of the total number of positions.
BTW, I recently noticed that in the last years, a lot of really interesting innovation is in the hardware space (Arduino, Internet of Things, Wearable computing, Osmo, Oculus Rift...). Before you walk away from an M.S. in EE, see if you just need to find your inspiration, and maybe think about hybrid soft+hardware roles that let you build on your education.