This is a two-part question. I'm going to pick on Craigslist, but I think it applies to other sites as well.
One of the drawbacks to free/anonymous posting is the amount of spam (including mass-posting across inappropriate categories), scams and phishing. This problem seems to be getting worse, and some sites have even disabled anonymous commenting to combat trolling.
Would a post/ad on a site that required registration with a real name (even if it wasn't visible) be more trustworthy? Would you be more inclined to trust an ad that was paid for (even a nominal fee)?
Second part: Do you think people would be willing to use such a site? Willing to pay a small fee to post ads? Or are people overall willing to put up with the garbage in exchange for anonymity/free? (Obviously people still use Craigslist, but if an alternative were available, would they use it)