The idea with exceptions is that you catch them only if you want to handle them and continue in some way. Usually this means it's your client code that would be catching them, not a base class, although it's possible that doing this in a base class could make sense in certain situations.

Converting them to error codes is a red flag because error codes have problems that exceptions were supposed to address -- namely that if you forget to check one, your program will continue as if everything is okay.