"I don't have no money," still sounds wrong to my ears, but give us a half millennium and I'll get used to it. In most languages, negatives don't cancel each other out, they just add emphasis. In Spanish, "No tengo nada," (I don't have nothing) just means you don't have anything. In English - for the time being - I don't have nothing, means you have a little something. English owes its unusual proscription against double negatives to 18th Century grammarians better versed in the strict rules of algebra than the fuzzy logic of grammar. Since in math, two negatives make a positive, they reckoned it was the same way in English.
The sad thing is, as we wave good-bye to yet another slowly-departing rule of English, welcoming the double-negative means we will lose litotes.
Litotes is a figure of speech that asserts a positive by negating its opposite. "I don't have no money," actually means I have a little: not enough for the bald statement, "I have some money," but too much for, "I don't have any money." That's litotes, and it hits a very nuanced little target of meaning. But once double negatives have their way, it's farewell litotes forever. A sentence like, "I'm not without influence," would be completely opaque to an English speaker with no sense of negatives' canceling each other out, and "I'm not without influence," is a lovely sentence, a charming sentence.
But after all, how important are litotes anyway, given the enormous convenience and simplicity of being able to rip off with a good double- or even triple-negative? "She didn't never love him none." It does the soul good to see it there.
Besides, do I ever really use litotes? Does anyone?
Well, I don't never use them.