First we need to define "Balkanizes". How many states are forming out of the US? Is it each of the 13 Colonies for themselves, or regional formations? Because regional formations will be much more stable and viable than each colony for themselves, which might as well lead to places like Delaware being conquered by Pennsylvania or something. Or perhaps an analogue to the Argentine Civil Wars with a Rosas-like figure in charge (I've always liked the idea of Andrew Jackson as the US's Rosas). You also have to worry about new states in the west, like the State of Franklin and such assuming the "Thirteen Republics" can't hold them down.
But let's go with regional federations for now.
Well IMO, it was the stability of North America that drew people to the USA, a balkanised US would mean less immigrants over all to the balkanised States, maybe more settlement in Canada, Brazil and Argentina. With less people I would expect less economic potential, meaning the combined gdp of the balkanised US will be less today then an intact United States.
Overall less people, less money. which could mean that North America would have a larger native population.
Why would it necessarily be unstable? If you make three states that are New England, Mid-Atlantic, and South, the South is the only one that might be unstable since it would be closest to a Latin American state, and even then, plenty of Latin American states were more or less stable in the 19th century, not to mention as a slave-based economy the South will have difficulties attracting immigration as it did OTL (not that it won't exist, of course).
I don't see how it follows the US would have a larger native population. Some groups would certainly be larger, since the Cherokee, Creek, etc. will continue their integration into society unless the South muscles the manpower to evict them to Oklahoma or elsewhere as OTL. But many other groups will still end up pushed aside and beaten. It wasn't even immigrants who were doing this (for the most part, Germans are the main exception), since immigrants tended to settle in cities--these were Anglo frontiersmen and pioneers who were at the forefront of the Indian Wars, and it was on their behalf the wars were fought.
I'd still expect New England to extensively attract immigrants, as would the Mid-Atlantic regional federation. It would certainly be weaker than the US as a whole, but the draw for immigration would only be slightly less, unless you can bring anti-immigrant hysteria to new levels and shut off immigration to one or both of these areas early. But remember that the situation in Brazil and Argentina when they received most of their immigration was far better than it was in the early 1800s when the US received much of its immigration. If a Balkanised US might be unstable, then early 19th century Argentina is gonna need to look significantly different for it to even compare stability-wise.
Now for the scenario I proposed above, I think there you'd see much less immigration, but it would still no doubt be attractive to immigrants. Frontiersmen in New York and Pennsylvania, parts of New England being like Belgium in terms of early industrial potential, etc. There's still lots of potential, and you'd see regional cooperation between some states (although not all of them) to increase the economic strength of the various states.