From a short note by Michael Covington, "Has Chomsky been blown out of the water?":
Chomsky’s first major contribution to linguistics was item 1 in the outline, mechanisms to describe syntax precisely. Before he came on the scene, linguists widely accepted the behaviorist dictum that science can only refer to observed behavior. Some took this so far as to forbid that the study of meaning (which is only observable within the mind), which greatly impeded the study of sentence structure. Chomsky argued cogently that abstract models are as appropriate in linguistics as in physics. He allowed syntactic theory to be abstract enough to do its job.
Besides the tree structures that are now familiar, Chomsky introduced transformations, which are rules that turn one tree structure into another. At first these were used to turn one kind of sentence into another, such as declaratives into questions. They made the grammar more concise; after you accounted for one kind of sentence, the transformation gave you another kind of sentence with no extra work.
Very soon, transformations took on a different function, to build observed sentences from more abstract structures that are not observed. This made it possible to simplify the grammar and make it more general.
Since Chomsky's early work linguists have developed many methods for precisely describing syntax, some inspired by Chomsky, some not. More than anyone else, he got that ball rolling.
What's in doubt is his postulation of universal grammar:
He proposed that the principles of grammar are the same in all languages; the brain is pre-programmed for universal grammar. To learn a particular language, all you have to learn is the vocabulary and a set of “parameters” that establish how the grammar rules play out in that particular language. That is how children manage to learn to talk so quickly and efficiently, without being taught, from hearing incomplete examples of speech. [...]
The second is advances in cognitive psychology. Back in 1957, there was almost no cognitive psychology. Chomsky was rebelling against Skinnerian behaviorism. Most psychologists at the time advocated an impossibly simple theory of how the mind works. Against that backdrop, he had to postulate inborn universal grammar in order to do everything that simple stimulus-response associations couldn’t do. Nowadays we know that the human mind has complex, powerful, abstract capabilities in many areas, not just language, and new possibilities are opening up for explaining language from general mechanisms of thinking and learning.
This judgment seems right to me:
What is pervasive is the expectation that other linguists will investigate questions that Chomsky raised, whether to prove him right, prove him wrong, or follow the evidence in some other direction entirely. That’s how science is done. This expectation is warranted because Chomsky has raised undeniably important questions.