They cannot destroy it but they can hinder it in major ways and drive it underground. For example if major governments announced that they're banning bitcoin it would drive off a lot of users and we would have to find ways around their legislation. Major exchanges will have to go underground, it would be a lot harder to buy bitcoin, maybe the value would drop.
What would 'banning bitcoin' even mean? All that is required is the computation of a number and its transmission to the blockchain. It's impossible to block such a message without blocking all other communications.
likely, "banning bitcoin" would take the form of barring companies from accepting it. It definitely changes the value proposition if Amazon, Target, Overstock, Subway, TigerDirect, Home Depot, etc. can no longer accept bitcoin as payment. There's no need to mess with the blockchain if you can simply eliminate mainstream use and force it onto black markets.
> On what basis could a company be banned from accepting it?
If by basis you mean authority, the Commerce Clause is most likely, though I suppose the Coinage Clause might be invoked as well.
If by basis you mean policy motivation, there's an infinite number of possibilities.
> No physical items are exchanged.
Government regulates acts that don't involve an exchange of "physical items", but just electronic data, all the time (pretty much all instance of wire fraud, CFAA violations, among many examples.)
Engineers like to use lines like "long chain of numbers", but that doesn't really hold weight. Stolen credit card numbers, digitized child porn, and terrorist communications all also qualify as "just a bunch of numbers". So does the majority of NSA surveillance, which most HN'ers oppose.
The government is pretty good at figuring out justifications for banning things. I wouldn't rely on reasoning as flimsy as "it's just numbers" for why it couldn't happen. (I'm not saying I expect them to, just that if they did, it'd most likely take the form of a merchant ban, "just numbers" would not be an adequate justification, and all the big retailers would comply because none of them want to risk a big legal fight.)
> What would 'banning bitcoin' even mean? All that is required is the computation of a number and its transmission to the blockchain. It's impossible to block such a message without blocking all other communications.
Government prohibition of an act rarely involves blocking all instances of the act (in fact, if the government could effectively make an act impossible, it wouldn't need to prohibit it.)
So the question here just seems to miss the entire idea of legal prohibitions.
Make it illegal? Break down our door and take your computer and coins? Raid the service that holds your coins? Put you in jail if you refuse to give up passwords?
You miss the point, they don't regulate Bitcoin, they regulate citizens. Just because you can't stop something completely, doesn't mean you can't outlaw it. Bitcoin, like all commodities now and in the future is taxable under current law because the law taxes all income of any form except things we say you can deduct. That means it covers new forms of assets as well as existing ones. Not paying your taxes is illegal and is more than enough threat—as it always has been—to make citizens obey the law.