These days, at least for the long-haul part of the path, most of the internet is fiber optic and probably not susceptable to solar flares. That's not to say that the "last mile" might not be susceptable. On the other hand, it's a well kept secret that the SDRAM in PC's is susceptable to cosmic rays. I saw a study a few months ago about how A) servers have what was once regarded as vast quantities of memory, increasing the odds of hits and B) the feature size of the memory cells is smaller, further increasing the likelyhood of a cosmic ray disrupting the contents. The gist of the article is that there's a real need for error correction memory in servers.
I'm not an astrophyscist (and I can't even spell it) so I don't know if solar storms produce cosmic rays.