I've found that no matter what I do, it continues. The first thing I did when it started doing this was to mark them listened to. I even tried "listening" to them -- playing them again, but skipping through them -- to see if that made a difference. With the 11.1 update(?), a 24-hour wait between listening and deleting was added, and it would delete them after 24 hours . . . and then immediately re-download them. Like, within the next couple of refreshes. I tried unchecking them and marking them played. I tried both of those + then deleting them. I tried unsubscribing and resubscribing. I tried unsubscribing, resubscribing, then immediately right-clicking the podcast itself and saying 'mark played.' (That's when it then unsubscribed me from the podcast, and I had to resubscribe.)
None of that worked in any way. It downloads at least six played episodes of one podcast including the most recent few and then a couple six to eight months old, several episodes of another one or two . . . and then, this morning when I got up, it had re-downloaded something like 200 episodes of a podcast I'm subscribed to, but I've listened to every episode. I marked all of those played, unchecked them, and deleted them. So far, it hasn't re-downloaded them, but the day isn't over. 🙂
I also tried going to the feed view, and marking them all read and unchecking them from there.
Also didn't work. I'm pretty sure that either I have done something very bad and corrupted a database OR there's a bug. Or I've managed to get the settings so messed up that it's in some mode . . . who knows.
I did speak with an Apple support person earlier tonight (on a holiday-weekend Saturday!), and I uploaded a log and some other files for them to sift through to see if they can identify what my problem is. I could not be more pleased with the interaction so far. I hope they find something, even if it's a loose nut at the keyboard.