To a certain extent, yes. Not every designer will be comfortable with, or have interest in, learning a skill that has distinct differences from print design. However, as technology continues to advance, and we turn to digital media as our main source of communication, web design will become an expected part of a graphic designer’s skill set. One of the main reasons I feel web design is an important skill, is purely from a marketing and self-publishing standpoint — having the ability to create your own web site, or to manipulate the CSS of a Wordpress, Blogger
or Tumblr site, is a valuable resource in the digital age.

To those print designers looking to get started in web design, I
would like to make a few recommendations. Always design at 100% screen resolution. Most print designers commonly zoom in to 400% – 1600% to deal in subtle detail; detail at this level does not exist in web
design. Spend time learning Dreamweaver and CSS. CSS has fantastic functionality, and once you “get it,” it’s not difficult to use, but it
does take practice. Also, for added interactive functionality, look into Javascript vs. Flash. Flash takes up larger amounts of memory, causes user issues if a visitor is not updated, and Flash is being systematically phased-out by Apple mobile products.

A few worthwhile resources to get you started understanding CSS are: w3schools.com, lynda.com (a subscription site with videos / tutorials) and csstypeset.com.