save system works fine, i tested the 3rd birthday / crisis core and saving works perfect, make sure you're using an official build (not emucr or some other source) they aren't supported., if you still have a problem set jpcsp logger console to info, compress the html file and post it in the game thread so the developers can look at the error.
Try Out JPCSP Launcher v1.8.0.4 | How to post a log