I don't think the EFF have a particularly anti-copyright position. I mean, EFF cofounder Mitch Kapor was the cofounder of Lotus, which definitely did not give its software away for free. It seems his main motivation in the early days was a worry that a computer-cracking backlash would go too far and violate civil liberties (e.g. in the infamous Steve Jackson Games case), which isn't the same as thinking that computer cracking is a good thing. The organization now has similar concerns about whether the anti-piracy backlash is sufficiently respecting due process, protecting the rights of people who are in fact innocent, etc.

Somewhat more true of Creative Commons, but they still have an opt-in view of it: they'd like to promote a world where many people give away their music for free, under various definitions of "free" (may include e.g. no-commercial-use clauses), but it's strictly voluntary if you want to use a CC license or not.