I haven't used SoundTap much lately, but I do remember when I was using an older version quite a bit (back @ six years ago on an xp system) opting to start a new recording after a pre-set silence was simple and worked well.
As I look at it now (I have version 2.11) I can't see any way to do this. So I can't offer much help, but if you are watching/listening while recording a stream, you can manually stop and start recording to separate files.
Anyway, I open the files in WavePad to edit; there you can split and trim (and much more) although it is a bit of labor probably not to everyone's liking I'm sure...