Basil is traditionally masculine. Someone could still name their daughter
Basil, rather like they could theoretically use
Michael or
Archibald, but it doesn't have a history of use as a name for girls, so it isn't unisex.
It is derived from "basileus" meaning "king", "emperor" or "tzar," which may be why it strikes me as a little pompous in an antiquarian way. It has that British boarding school vibe as well.
Famous people named
Basil that come to my mind are the British actor
Basil Rathbone (known for portraying
Sherlock Holmes), and the "Great Mouse Detective." There were also a couple of Byzantine Emperors named
Basil.