If you're having an issue playing Borderlands 2 DLCs and are receiving a message indicating that you cannot play the DLC without an internet connection, this is likely due to what's called "Roaming".
The term “roaming” refers to using content that you purchased on one Xbox 360 console on another Xbox 360 console. For example, you may download a game from Xbox LIVE Games store to your Xbox 360 console. You may then take the hard drive that contains that game and attach the drive to another console. When you attach the hard drive and sign in to Xbox LIVE, you can play the game without any restrictions.
Note For such roaming to be allowed, you must be connected to Xbox LIVE by using the profile that purchased the content so that the usage rights can be verified.
For more information, please refer to the following Knowledge Base article by Microsoft Support: