The current JLA's heaviest hitters are Hal, Donna Troy, Dr. Light (the hero one of course), and Starfire. So... they aren't really set up to take on Thanos-level threats.
The JSA, well, they split into two teams recently. The ones calling themselves the JSA do have Dr. Fate, and Obsidian, but I think Alan Scott's out of action and several of the other heavy hitters are over on the All Stars. So, Fate and Obsidian.
So they're rather light on the needed muscle at the moment.
Jakeem's not exactly active in the team recently, and yea, I forgot about Obsidian being an egg and Mon'el in (though Obsidian would've been far more useful than Mon).
Hm, if one included the All-Stars, that'd add some numbers, but... with Magog off, Power Girl's the main addition there.
Yea, right now DC's big teams are not in a big position to take on cosmic threats.