Both formats are correct. Saying "I don't have a..." is used to talk about a singular object. So if someone asks if you have a pen that he could borrow, the most common response would be, "I don't have a pen." Although you could say "I don't have any pens" if you want to emphasize that you do not own a single pen - not with you, not at home, not at your office, etc.

Regarding the sentence referring to money I personally feel that "I don't have any money" is the most common response. However the connotation could be different in many different scenarios depending on the context in which it is used.

Hope this helps!

Best,

In Perfect English