(Alternatively, if they were to publish them, would the US Government be forced to respond differently to how they did with WikiLeaks, so as to not be seen as hindering free press? I'm sure in the public's eyes, going after WikiLeaks is a much smaller thing than going after a paper as big as WSJ.)
Of course not. Wikileaks isn't solving a technical problem. They need to exist because traditional news agencies aren't willing to take the hit associated with releasing unflattering information. We saw what happened to financial businesses that dared to provide perfectly legal services to Wikileaks. Journalists who imitate Wikileaks can expect to be harassed, fired, or even prosecuted.
I find it ironic that so many traditional media figures bemoaned the loss of their "gatekeeper" role when the birther controversy was still monopolizing headlines. They have no trouble acting as a gatekeeper when journalism threatens the rich and powerful.
In short: if publishing doesn't align with their best interests, they can be relied upon to not publish.
Well the US Government has yet to manage much in the way of actually going after wikileaks. The Wall St Journal has the advantage of falling under its jurisdiction.
But the answer is: no, I get the impression that the WSJ will be more circumspect about precisely what it publishes. On the other hand if the WSJ does publish your leaked documents they automatically get a lot of credibility. And if you leak documents to the WSJ and they refuse to publish you can always leak 'em to someone else.
On the other hand, it's not like Wikileaks is publishing things indiscriminately nowadays -- they're still sitting on a huge stash of diplomatic cables and releasing a few every time they feel like it. As I've said before, if I were the wikimedia foundation I'd be complaining pretty hard about the fact that Wikileaks isn't actually a Wiki any more.
But, PR can spin WikiLeaks as anti-American, it's much harder to do that while trying to shut up the WSJ.
I don't imagine much fun information coming out of this.
It allows them to sue the anonymous uploader for breach of contract or perjury or whatever if they want (which they don't), but it turns it the rights into the uploader's issue, not Dow Jones'.