Is "Save downloaded subtitles to" selected under [Online subtitles]?
yes OK
If playing from read-only media: did you (for this purpose) define a custom (fixed) subs-folder on your HD?
I don't know what you mean here (1) Do you play your movies from your HD or from (e.g.) a CD? (2) What is selected in the white box at "Save downloaded subtitles to"? NB: If you play the movie from read-only media (like e.g. a CD), BS.Player will - of course - be unable to save the downloaded subs in "movie-folder" (or a subfolder 'Subs' thereof). You should then define a "fixed" folder to save the downloaded subs on any of your HDs. E.g. F:\BSplayer\Subtitles. (NB: also as "Subtitles folder" under [Subtitles properties])? The "Subtitles folder(s)" defined here will be searched for a sub. If the above applies F:\BSplayer\Subtitles should be a (one of the) folder(s) defined here.
Is "Download only if local subtitles are not available" selected?
yes OK
Is "Remember movie settings" selected under [General settings]?
no If you set this to "yes" BS.Player will remember i.a. subtitles for each movie played. 
(In this connection pse note that all online subtitle-databases are dynamic.)