To back up your saves (which is a good idea, because of Stsrflight's unique approach to saving game progress), you need to make copies of these files:
They are in the game's directory - by default, that will be:
C:\GOG Games\Starflight 1 and 2\Starflight 1\STARFLT\PLAY
To find out how to back up gamestates on Mac OS X, click here.
f you ever quit the game without saving and get the "Game in progress not saved" message, you need to open the game's Start Menu folder and launch it by using the "Reset & Start Game" link. On some systems, you may have to launch this option by right-clicking and selecting "Run as Administrator", in oder for it to work properly.