Should an employer care about their employees financial problems?

Yes and no

I would only care if it affected their performance. 

Another concern might be if they handled cash.

Actually, it makes some sense--especially in certain  occupations. If an employee is having money problems, and they work in a bank, the temptation to take cash out of the vault might be an issue. If creditors or garnishments are constantly contacting the employer, it's also his problem as well as the employee's. 

But just in general? It's not his business. Most empolyers won't ask about it, unless there are issues that arise because of it. 

Up to a point.  If someone asks to change to two fulltime days instead of five part time days on a minimum wage job it's worth considering that the savings in bus fare will put substantially more money in the employee's pocket.  I would try to make that happen.  At the other end of the scale in government and finance someone with financial problems would be considered to be high risk because they can be bought or they may be tempted to embezzle.

No. Meddling in any employee's personal life is really not acceptable behavior. It is also bad form for an employee to complain about their personal problems, including financial problems, at work as it is your private business.

IF you do an exceptionally good job & feel you want to apply for promotion, that's a different matter. So is asking for a raise if not offered at your annual performance review. But you don't use your personal financial problems as a reason to ask for a raise - you convince your employer your performance & value to the company merits it. You can also ask for or volunteer for extra hours if you are an hourly-wage employee, and that way try to earn more.

 But you really need to manage your finances more effectively. You do not want your employer to consider you incompetent in any way, including managing money. If you cannot manage your money, your employer cannot rely on you to manage company resources effectively.

An employer is focused on keeping the business running,  not about everyone's home life. 

Yes and anything else, not that you have a magic wand but you can point them in the right direction if you can't help. My employer has a personnel officer you can go to for anything at all. If it wasn't for employees then the bosses wouldn't be where they are.

Absolutely.  I wouldn't hire someone who is incapable of handling their personal finances.  If something beyond their control is causing an employee financial problems (family sickness, house burning down, etc.), I care and want to help.