How critical is it if SYSAUX is full?
If your Database has enabled AUDIT and your auditing tablespace SYSAUX gets 100% full,  NONE of database users will be able to log in to database anymore. Your database will not crash but only SYS db user will be able to log in - the first thing you should do in this case is to extend the SYSAUX tablespace (add a data file) and deal later with how SYSAUX got full at the first place.