Power level: reasonably high. This will see widespread play in brain damage decks, particularly those using Thule. It extends the game, giving you more time to set up your win conditions and also helps charge up Stock Buybacks.
While the intended recipient of this is almost certainly HB brain-damage decks, even at 4 influence it will likely see some play in Jinteki PE decks that don't have many alternatives for either extending the game or invisibly guarding Archives. Many Jinteki threats require or benefit from facedown cards in Archives so the runner probably does need to check Jinteki archives more than normal. Getting to 7 agenda points against a PE grinder is mentally taxing. Playing to (say) 9 points or on a max hand size of 3-4 is brutal.
The problem here isn't the power level, but the anti-fun experience of negative agenda points. In a card pool where 2 brain damage is crippling against like 10+ card slots in PE decks and potentially a win condition for Ontological Dependence players, giving out brain damage or negative agenda points for free on an access anywhere doesn't feel like it's on the right track. I think Global Food Initiative is a less frustrating card to interact with, there's still some forward progress when you steal it, just not as much as normal. Nightmare Archive might have been more fun to interact with if it had been tied to forward progress, perhaps a 5/3 agenda which inflicts a point of brain damage when stolen.