Martians attack from outer space. Until the Martians really attack (which we're expecting around 2025, so mark your calendar), that counts as science fiction for us. Like the best science fiction, The War of the Worlds confronts us (or at least its original readers) with something new and unexpected – not just Martians from outer space, but the idea that the British might be colonized by more advanced beings. (As opposed to the idea that many of the British probably held at the time that they were the advanced beings and should be doing the colonizing.)
Well, in The War of the Worlds there's danger and a lot of running and hiding, which we might consider a component of adventure. We were tempted to mark this down as a Quest, but really, a quest should have some positive goal – a Holy Grail or dumping a ring into Mount Doom or something like that. While the narrator occasionally talks about how he's trying to find his wife, he doesn't really do such a great job of trying to get to her. Mostly he just runs away from the Martians. (The narrator's brother has a more standard quest, to get to the shore and save the women.) In that case, let's just call this an adventure.
But what a terrible adventure story. Usually, in an adventure, we want an active protagonist who will fight back. Instead, what we get in this story is a guy who runs around, hides, and does a lot of peeping through a little hole.
The narrator isn't a soldier and he only experiences the war as a refugee, but we might still consider this novel a war drama in part. After all, the refugee's story is still a story of the war. Now, the biggest genre of this novel is still science fiction – after all, the enemy army is from Mars – but we can see how this story does use war as central issue.
Lastly, the Martians are monsters through most of the text. They are horrifyingly different from humans, they suck blood, and they are unstoppable. That sounds like a nightmare, so we could read this text partly as a horror story. There's an interesting connection to be made between The War of the Worlds and Dracula since both feature blood-sucking invaders. The only problem is that we usually expect some heroic protagonist to fight off the invader. That's what we have in Dracula, but not in The War of the Worlds.