Ain't, in I ain't got no money means have not. With that meaning, it originally represented the London dialect, which uses sentences such as they ain't got nothing to say.
About the usage of ain't, the NOAD has the following notes:
The use of ain't was widespread in the 18th century and is still perfectly normal in many dialects and informal contexts in both North America and Britain. Today, however, it does not form part of standard English and should not be used in formal contexts.
The sentence then contains a double negative, but the double negative is sometimes used to give emphasis to a negative.
The NOAD has the following notes, about the usage of double negatives:
According to standard English grammar, a double negative used to express a single negative, such as I don't know nothing (rather than I don't know anything), is incorrect. The rules dictate that the two negative elements cancel each other out to give an affirmative statement, so that, logically, I don't know nothing means I know something. In practice, this sort of double negative is widespread in dialect and nonstandard usage and rarely causes confusion about the intended meaning. Double negatives are standard in other languages such as Spanish[, Italian] and Polish, and they have not always been unacceptable in English. They were normal in Old English and Middle English and did not come to be frowned upon until some time after the 16th century. The double negative can be used in speech or in written dialogue for emphasis or other rhetorical effects. Such constructions as has not gone unnoticed or not wholly unpersuasive may be useful for making a point through understatement, but the double negative should be used judiciously because it may cause confusion or annoy the reader.