They were most likely music stands or decorative stand fronts used by big bands to hold sheet music and make the band look more polished onstage.

In the big band era, these could also be called:

  • Band fronts or music stand fronts.
  • Stand covers used to hide the plain stand and sometimes storage underneath.
  • In some performance settings, a raised platform for the band itself would be a bandstand.

A quick clue: if you mean the individual objects in front of each player, it’s almost certainly the music stands. If you mean the whole raised area where the band performed, it’s the bandstand.

The big band setup itself usually placed saxophones in front, with trombones and trumpets behind them, so the stands or fronts would line up across the section rows.