Now we've got more definitions confused than we should. Let me quote Wikipedia:
Anti-intellectualism is hostility towards and mistrust of intellect, intellectuals, and intellectual pursuits, usually expressed as the derision of education, philosophy, literature, art, and science, as impractical and contemptible.
How willful it is is a question that must be asked first towards similar beliefs, such as racism and misogyny: do people take on those, or do they imbibe them as their own from their surrounding? From the answer to that question stems the answer to whether it is, or can be, self-aware or prideful, and from those - whether it is ignorance (willful ignorance, in this case).
Whether it is, people still hold onto them without giving them a good critical look. That constitutes ignorance in my book.