I tend to agree with Martin - it seems like there's probably a way to preserve some of the power of the old site and 301-redirect selectively (or potentially use cross-domain rel=canonical tags), but it would take a much deeper understanding of the site than Q&A allows.
If you rebuild the site from scratch, you'd almost always want to de-index the old site. I'd flat out remove it via Google Webmaster Tools - it's the fastest method. Leaving both sites crawlable is only going to compound your problems and haunt the new site.
I'd warn, though, that if this is Panda-related, just moving the content won't solve your problems. You do have to sort out why they happened in the first place, or the same algorithmic issues will just come back. In other words, if the problems are content-related, then it doesn't really matter where the content lives. If the problems are link related, then moving will remove the problems. Of course, moving will also remove and advantages you currently have based on good links.
Unfortunately, this isn't a problem that can be addressed without a pretty deep audit. My gut feeling is that there may be a way to preserve some of the authority of the old site, but you really need to pin down the problems. Panda + Penguin is a wide swath of potential problems and just isn't enough information to do this right.