There does not seem to be much of a demand for pure web designers these days. Most designers these days need some experience with programming languages in order to integrate their work with developers. In the last 5 years, I've only met one person that only did pure design.
In regards to your question of "high-paying", that will depend on your definition of high pay. In a recent article in Money magazine, software developers ranked #1 in an article about best jobs. It was a combination of job growth, personal satisfaction, flexibility and salary.
If design is your passion, I would load up on skills in the area of web-design and focus on tools such as Adobe Photoshop, Flash and HTML. I would also familiarize yourself with some of the more popular web-languages such as Java and C#. Also familiarize yourself with how designers work with developers. There are techniques that prevent the two from conflicting with each others work.
Good Luck!
Vince