Browsers should actually issue patches that look at the list of affected sites, and looks at the cert dates, and pops a warning that it's not secure.
If all major browser makers do this, it will force these sites to fix the issue fast because they'll be getting tons of calls that they're getting an error.
Basically the browser looks at an online list of a known affected site and when it was patched, if that info is available. Then it looks at the site's cert to see if it's been updated after the patch date.
Not 100% fool proof and it has to trust that the data is real (ex: if the company said they patched, trust that they really did).
I have a feeling this bug will get forgotten in a few weeks from now after enough stabbings and shootings have happened to take over the news, and lot of companies wont bother to patch or reissue certs or do whatever they need to do to fix it.
Not that we could trust anything before knowing what the NSA does, but right now, even more so, we really can't trust that what we're doing online is actually encrypted, since we don't know what these sites have done, if they redid their certs etc...
Online banking is probably the biggest one. NSA will just pass all your banking info (transactions etc) to the IRS and they'll have a field day with that.