Rationalism in philosophy is generally contrasted with empiricism. I would say you're a little off in characterizing anti-rationalism as holding rationality per se in low regard. To put it very briefly: the Ancient Greeks set the agenda for Western philosophy, for the most part: what is truth? What is real? What is good and virtuous? Plato and his teacher/character Socrates are the archetype rationalists, who believed that these questions were best answered through careful reasoning. Think of Plato's allegory of the cave: the world of appearances and of common sense is illusory, degenerate, ephemeral. Pure reason, as done by philosophers, was a means of transcendent insight into these questions.

"Empiricism" is a term for philosophical movements (epitomized in early modern British Empiricists like Hume) that emphasized that truths are learned not by reasoning, but by learning from experience. So the matter is not "is rationality good?" but more: what is rationality or reason operating upon? Sense experiences? Or purely _a priori_, conceptual, or formal structures? The uncharitable gloss on rationalism is that rationalists hold that every substantive philosophical question can be answered while sitting in your armchair and thinking really hard.

Well empiricists think knowledge exists in the environment and is absorbed directly through the eyes and ears without interpretation, if we're being uncharitable.