What cooperative board games are there that I could safely play with my wife?
Arkham Horror is almost completely cooperative.

It is the players against a Great Old One(GOO) of Cthulhu mythos.  Players need to work together to arm themselves and find clues that will enable them to seal gates that open from other realms.  There is a large number of expansions, but the basic game is very enjoyable.

Generally, the expansions make the game harder and should be avoided until you can regularly beat the GOO in the base game.

See Is it possible to play 2-player, 2-character Arkham Horror successfully? for a question about the viability of a popular 2 player variant for Arkham Horror.