I'm not a Swift user myself, so I'm not sure if there's a better way to do it, but I believe you need to create a "bridging header" to use C framework/library headers. For example, create a "bridging-header.h" file containing this:
#include "bass.h"
// #include any add-on headers here too...
And then set "Objective-C Bridging Header" to "bridging-header.h" in your project's settings. The BASS header(s) should be placed alongside it (or somewhere in the header search path).
Btw, with the switch to dynamic libraries in BASS 2.4.17, any dependency frameworks can be (and are) specified in the BASS libraries themselves, so there's no need to include them to your project's settings any more.