I know anti-government fanatics will oppose this no matter its merits, so I wonder why I'm writing this post. Still, here goes.
Many European countries have been using electronic ID cards (yes, we have those and we aren't being oppressed by the evil government yet) that also serve as an internet ID like the one proposed by Obama.
So far there are no reports of it being exploited, hacked or otherwise used for malicious purposes. On the contrary, it has reduced preying by paedophiles (as many online chatrooms attended by young users now require you to use such an ID, thus preventing you from lying about your age), has increased the security of online banking and so on.
The costs are extremely low (we get readers and ID cards for free, but even if you had to purchase them, costs are just a few dollars), the security improvements massive and the amount of private data stored minimal (the government can hardly use your age, area of residence and other such basic facts to oppress you in anyway, as they already have these anyway)