Yes, you should be able to play cooperatively with your friend, bearing in mind that lacking the DLC, your friend probably can't join you in any DLC areas (Tiny Tina's, Mr. Torgue's, Sir Hammerlock's, Captain Scarlet).
However, any paid characters like Gaige or Psyco should not interfere - My friend has no issues connecting with me as I play mechromancer, while he owns neither, and I've played with plenty of Psyco's while having not purchased him.
I am, however, a little foggy on how some things might work out; like Ultimate Vault Hunter Mode, with your DLC-lacking friend.
Additionally, your friend won't be able to use any DLC weapons that you may want to trade to him.