To be honest, in my experience, a degree hardly matters for web development, and it certainly doesn't matter what you major in.
I put in the time -- and money -- to get a bachelors in computer science, but when I was hired to a corporate-style job, the degree mattered only to avoid any automated exclusion filtering, and when I was last hired to an agency-style job, I didn't even bother to mention my degree at all; they only care what skills you have.
There's also a huge gap between academic web development and professional web development. Four years and tens of thousands of dollars will nonetheless leave you unprepared to do any real-world work.
But there's good news too. It used to be that you needed to go to the most prestigious schools if you wanted to learn from the smartest people in the field. But today, you can learn from the smartest people in the field through [blogs, [URL="http://www.youtube.com/playlist?list=PL5586336C26BDB324"]YouTube, [URL="http://twitter.com/dalmaer"]Twitter, [URL="https://developer.mozilla.org/en-US/learn"]etc](http://fabien.potencier.org/).
Finally, here's the best advice I can give you. Scope out the kinds of places where you'd like to work and get in touch. Ask the employers directly what kind of background they look for, what kind of skills they look for, and get their advice on how best to learn those skills.