Number of subsets on a set with partial order
See three closely related questions Does faster exact algorithm for counting independent sets in comparability graphs than general graph exisits? and Is counting maximal cliques in an incomparability graph #P-complete? and https://mathoverflow.net/questions/49515/on-the-number-of-antichains-of-a-poset

They are all on antichains rather than downward-closed sets but downward-closed sets correspond one-for-one with the antichains of their maximal elements.

I believe both of the cstheory questions have answers with references that also answer your question: no, the problem is #P-complete.