Card interaction: Basilisk and draw then discard.
There's a comprehensive FAQ compiled from answers to questions from the game designer

Q: If a Villain blocks part of a card effect, can you or must you execute the rest of the action?
A: Yes.  For example, Lockhart says “Draw one card, then discard any one card.”  If the Basilisk is in play and you choose to play Lockhart, you would not draw the card but you would still have to discard.  Neville could thus use Lockhart to discard his Remembrall to gain two influence.
Source: https://www.boardgamegeek.com/article/24479506#24479506