We use the two expressions slightly differently. Your car keeps breaking down and your friend says "why don't you buy a new one"? "Because I haven't got any money" - of course you probably do have some, but not enough to buy a car. If you say "I don't have any money " it is more likely to mean I don't have any money with me right now (but you do have some money available - perhaps at home, or if you went to the bank). Whichever version you use, you need the 'any' in there between 'have' and 'money' for it to sound natural.

"I don't have any money" or "I haven't got any money" have pretty much the same meaning. I haven't got any money could be used if you wanted to make the phrase a little bit stronger. (For example, if your friend says "Give me your money right now!" you might reply "I haven't GOT any money!"

The first one is better.