Can I invite friends to join me in the Crucible if they don’t have the dlc?
No.
In the same way, you cannot invite them to join you in the main campaign if they don’t have the base game. 
Can I invite friends to join me in the Crucible if they don’t have the dlc?
No.
In the same way, you cannot invite them to join you in the main campaign if they don’t have the base game.