So far, I've read or skimmed or at least glanced at 16 of the anthologies that my father owned. Here are some gender stats. Nothing particularly unexpected; this is just a recordkeeping/data-summary post.
Of those anthologies, eleven were science fiction and/or fantasy anthologies that were originally published from 1953 through 1982. None of those eleven include more than one story written by a woman. Five of them include no stories written by women.
The three post-1982 sf anthologies have slightly better numbers: 1984, 2 women (18% of the stories, in an Analog anthology); 1988, 2 women (13%); 1998, 3 women (14%).
I don't know how representative any of that is, in terms of all sf anthologies that were published during those periods; the novels my father owned were heavily weighted toward male authors, so his anthologies may have been too. But it certainly matches my perceptions; I've seen a whole lot of other sf anthologies that contained only one story by a woman. (Usually Le Guin, Zenna Henderson, or C. L. Moore, depending on the time period.)
(There are two non-sf anthologies: one, from 1954, has 42% stories by women; the other, from 1965, has 12%.)
I'll keep tracking this as I go through the rest of his books, but I don't expect there to be much change.