Depends entirely on the character. Daredevil, Black Widow, the Punisher, Spider-Man, Jessica Jones, Luke Cage -- street level and personal. Doctor Strange, Scarlet Witch, Loki, Iron Fist -- multiverses with huge stakes.
If you can get me invested in the character then I am always going to care what happens to them, whether they are on the streets of NYC or somewhere out in the multiverse. So no matter the size of the threat, if I really care about the characters, I still going to care about what they are experiencing.
I tend to prefer content to stay away from universal threats, cuz that can get old real fast. However, as other people have pointed out, it doesn't really matter to me as long as I care about the characters.
