It has just been announced when the Battleborn beta will begin. It will start on April 8th for PS4 and then the 13th for PC and Xbox One. The beta will end April 18th at 7 AM PST. Pre-loading for the beta will begin on April 6th and the download sizes for the beta:
PS4: 8.5GB
PC: 12GB
Xbox One: 11GB
If you play the PS4 beta you will receive the first Battleborn DLC Pack  and instant access to Alani upon release. Now only PS4 players will have access to the Battleborn hero Toby during the Open Beta.
The Battleborn Open beta will be available through PlayStation Network, Steam, and the Microsoft Store. You will not need any special code since it is an open beta.
Xbox Live Gold members will have access to all Story and Competitive Modes in the Open Beta. Xbox Live Silver members will be able to play private Story Mode episodes and private Competitive Modes, but will not have access to public matchmaking features.
In the beta you’ll experience two Story Mode episodes, two Competitive Multiplayer Modes, 25 heroes. The two episodes are:

• The Algorithm: Take on The Algorithm mission to debug and unlock the rogue Magnus AI, ISIC. If you played in the Closed Technical Test last year, you’ll notice Geoff, the Spider King, has made a few changes to his lair.
• Void’s Edge: Bring a jacket, ‘cause the ice-planet, Bliss, is mighty cold. A Varelsi portal has opened and it’s up to you to fight your way through the invasion and deliver the payload.

Also competitive multiplayer modes:
 

• Incursion: (Overgrowth map) Destroy the enemy’s two Spider Sentry Drones before they get to yours.
• Meltdown: (Paradise map) Send waves of minions to their death and score points for every minion who throws themselves into the incinerator. The team with the most scrap metal wins!