There’s nothing “wrong” with using pixels in an otherwise responsive layout, but if you do you’ll likely end up writing more code than you would using flexible units.
Jon Allsopp’s A Dao of Web Design predates responsive design by a decade, but its prescient advice remains perhaps the best way to approach any design, responsive or otherwise: “It is the nature of the web to be flexible, and it should be our role as designers and developers to embrace this flexibility, and produce pages which, by being flexible, are accessible to all.”
More than just embracing the nature of the medium, building your sites atop what developer Trent Walton calls “Flexible Foundations” can go a long way to making development easier. As Walton points out in his post, using pixels often means more code since pixel-based type, margins and padding mean you need to add new values at every responsive breakpoint.
“In many ways,” writes Walton, “every time we use a pixel value in CSS we’re rasterizing what was a fully-scalable web.”