Origin and correctness of “ain’t no”?

I had a linguistics class a long time ago where we talked about Black English Vernacular, with its own patterns and forms, and I think the patterns you're asking about are related to this discussion. Here's the first thing I found that seems to cover the bases, but I bet Wikipedia probably has a solid discussion on this as well:

http://www.linguisticsociety.org/content/what-ebonics-african-american-english

It's worth keeping in mind that double negatives are not universally grammatically incorrect and also appear as alternately positive or negative depending on the language. For instance, in Russian or Spanish, double negation is negation, while in English it emerges as its opposite, a positive. Thus, if you follow the argument that Black Vernacular English emerged from English mixed with west African languages or Caribbean creoles, double negation is probably something that rode along with the grammar of those languages.

What further complicates your example, probably, is the controversial "ain't". I am certainly no expert, though, and what I'm remembering is a discussion about twelve years old: maybe someone who has studied Black English Vernacular can offer an opinion?

I find it interesting that you point out that the double negative is incorrect, but you make no mention of whether "ain't" is acceptable or not. "Ain't" has an interesting history, and there is some basis for believing that it probably should have an accepted place in "proper" speech, but it has long been the focus of debate and controversy in the area of what is acceptable English.

If we go ahead and ignore the debate over "ain't," and grant for the sake of argument that it might have some valid usage, then we can move on to your main question, which asks why the double negative that you describe is becoming more common.

My feeling about the "origin" of this type of speech is that it arises from intrinsic human impulses. People often want to do two things in common speech: 1. Emphasize, and 2. Show "coolness." When you say "ain't got no," it is seen as being especially emphatic. It is more powerful than "I don't have any," which can seem effete. "Ain't got no" means "I'm telling you I really for sure DON'T have any." As for showing "coolness," this type of speech is considered to arise from the speech of those who show no affectation, those who prefer to say what people understand rather than what the academics would have us say, those who sound "hip" because of their relaxed mode of speech; all of this translates to being cool rather than being stuffy, overly formal, or aloof. When you speak this way, you're not "putting on airs" or acting superior; instead, you're "one of us" (regular folks).

The prestige dialects of American English (and British English) would indeed look upon "ain't no" as bad English.

However, there are common dialects in the USA where that is in fact the correct construction. The most prominent today would be African-American Vernacular English:

Negatives are formed differently from standard American English:[50]

• Use of ain't as a general negative indicator. As in other dialects, it can be used where Standard English would use am not, isn't, aren't, haven't and hasn't. However, in marked contrast to other varieties of English in the U.S., some speakers of AAVE also use ain't instead of don't, doesn't, or didn't (e.g., I ain't know that).[51] Ain't had its origins in common English, but became increasingly stigmatized since the 19th century. See also amn't.
• Negative concord, popularly called "double negation", as in I didn't go nowhere; if the sentence is negative, all negatable forms are negated. This contrasts with Standard English, where a double negative is considered incorrect to mean anything other than a positive (although this wasn't always so; see double negative).
• In a negative construction, an indefinite pronoun such as nobody or nothing can be inverted with the negative verb particle for emphasis (e.g. Don't nobody know the answer, Ain't nothing going on.)

Southern American English (which is closely related to AAVE) also uses negative concord, as do several British English dialects.