I realise that nowadays this would mean almost everyone, so the word wouldn't be used much. But in the past owning a thousand dollars would have been more notable so perhaps there would have been a word in common usage.
I don't know of any such word, and I would expect that it doesn't exist: nevermind the currency, 1000 is just not that large a number. It's only 10 times bigger than 100, so even if you've never seen 1000 of something all in big pile, you can easily imagine what it would look like - it'd be 10 piles of a 100 things.
A million, on the other hand, is so large a number that our everyday experience simply doesn't give us a good handle on it: it's a thousand thousands, and that's an awful lot of piles. Hence the desire to categorize those who hold that much wealth as something special, something worthy of its own word (i.e. apart from "wealthy").1
I think that even if you find a time, place, and currency in which 1000 represented a lot of wealth2, the number itself wouldn't be remarkable enough to warrant coining a word for it.
1An interesting side note is that in places where the currency has inflated to the point where a million is an everyday occurrence, "millionaire" still means someone wealthy: the term has become divorced from the number.
2Hint: 100 years won't do it, neither with dollars nor with pounds.
Actually, there is! And it isn't a humorous coinage, believe it or not.
From the Oxford English Dictionary:
n. thousandaire (after [millionaire])
One who has a thousand pounds.
First recorded use: The Eclectic Magazine of Foreign Literature, Science, and Art, 1896:
...to prevent their possessor from ever becoming even a thousandaire.
Note that I have bolded "pounds" in the definition. The Eclectic Mag. was a British magazine, and since the entry is marked as 'apparently an isolated use', it means that the OED can only say it is used in the context of the GBP.
However, this may not be the case:
Somebody whose wealth is greater than one thousand dollars, or the local currency. (emph. mine)YourDictionary
Origin: Modeled on existing words such as millionaire using thousand.
The fact that distinguishes a millionaire from someone who isn't is that the millionaire has capital. He doesn't have to work for money but he can live on the return of his capital.
The terms bourgeoisie or middle class both refer to people who do own more than the clothes that they carry. They own some assets but who don't own enough to life from returns of their assets.
Neither millionaire nor billionaire, reflect actual cash in hand, but rather to the net wealth of the individual.
The common usage word for a 'thousanaire' is 'poor'