UX expert Josh Seiden recently posted his affirmative thoughts about this question. I wanted to follow up with some thoughts of my own.
To me, this question is like asking should a writer know about the publishing business. If you’re “web-designing” solely as an artistic, self-directed endeavor, then “no” (but then we might prefer the term web art as opposed to web design). If you’re web-designing as part of anything larger, then “as much as possible, yes.”
As a developer and (novice) designer, I strive for pragmatism in both pursuits. Having both skillsets supports this. It’s pretty self-evident that the more of a system one understands, the more concerns and constraints they can simultaneously address/balance when working in that system. Understanding the relationship between CSS and HTML, their document-oriented roots, and the interpretation differences between browsers, better allows one to create flexible, idiomatic, conceptual, usable (e.g. performant) CSS while also supporting accessibility, SEO, and DOM and server performance.
I’ve read that the shift toward web design minimalism is about respecting the user. I agree, but I argue that it’s also about respecting the development process. Complex or pixel-perfect designs are very fragile and costly, wear on the development team, and hinder the business, usually with no marginal value over simpler designs.
Designing for web with strong knowledge of CSS is respectful of the process and environment of that work.