BSG Exodus: What happens with a damaged base star on launch raiders?
Nothing happens. The Exodus rule regarding raiders launching states:


  Launch Raiders or Activate Cylon Basestars: When either of these icons
  is resolved, if there are no basestars on the main game board, place 1
  basestar on the Cylon Fleet game board, according to the placement
  rules below, and advance the Cylon pursuit marker one space on the
  Pursuit track


Adding to the Cylon Fleet game board has an explicit condition which is not met in your example: no basestars on the main game board.