My mother is thinking about taking public speech classes now. Is it probable to improve my mother’s delicate anxieties in crowds due to mature age through public speaking classes? Will these better help make her cope with her fears?

Well the short answer is Yes, I can tell you there are many ways to deal with fears one of them is facing them. This is your case, public speaking is a good skill to have. It helps internally and externally.  

Just keep in mind nothing bad is going to happen, you are going to face critiques, but who cares, everybody receives critiques all time, so don't pay attention to that and support your mother to go ahead.