What type of joystick is it? Does it have all the same buttons as the standard Dual Shock controller? If so, it's just a 3rd party replacement controller & should work on all the PS1 games. It's possible that the one you have is just worn out if it's not working.
We have a Dual Shock 2 (the PS2 standard controller) hooked into our PS1 & it has worked fine on all the games I've played. You could just pick one of those up.