I would say that any good web developer should know how caching works - not just in browsers, but also proxies, firewalls, and every other link between your language parser (which may also cache) and the user's browser. It's "domain knowledge".
i consider knowledge of server caching is important thing a web developer need to know. i hardly doubt the knowledge of how browser cache is need to web developer. that's where the person comes in place called fronted developer or UI developer
A web developer needs to understand the entire chain of technology, from his text editor or IDE all the way to the browser. You don't need to be able to build a server from scratch (although you probably should be able to), but you need to understand how the pieces fit together (WSGI, mod_php, PHP_FPM, CGI, for example) and that includes which pieces of that chain can cache, might cache, will cache, and how they cache.
The front-end developer needs to be more concerned with browser-related issues, but doesn't need to know anything about the server stack (typically). The back-end developer likewise doesn't need to concern him/herself with browser issues, but needs a solid grasp of everything behind the browser.
A web developer has feet in both worlds and needs to understand much more than the front-end and back-end developers.
As an analogy, a gynecologist might be a front-end developer. He has a basic understanding of the entire human body, but specialized in this one area and everything related to it. People like to hang around him, hoping something will rub off.
A neurosurgeon might be more of a back-end developer (not what you thought, eh?). He's concerned with the code running things and how it will react with everything else. He can be hard to talk to, but always interesting.
The dermatologist is your UX guy. Presentation is his game, and nothing else (but he still understands the pieces that interact with his domain). His home is always awesomely decorated, but not always functional or obvious (the three shells on the toilet).
A general surgeon is the web developer. Solid understanding of everything, great at finding and solving problems, but knows when to leave something to the specialists. He lives in the home you wish you had, and his pool table always has great action.
I got carried away with that, but the tl;dr is: the web developer needs to understand the entire chain of his/her domain. There is no piece too small that you don't need to know something about it - like how it works, or how to fix common problems with it. From editor to browser, you need to understand how it all fits together.
