Examples: 1) Totally random site. Something I discover in a search or from a twitter link. Good design = engagement. Bad design = lack of trust and general skepticism
2) Less random site. Many people I "know" are using it. Something like gplus.to (or whatever that Google+ vanity URL site is). Good design = these guys must really know what they are doing. Bad design = this is probably one guys personal project blown up. It might implode so don't get addicted to it, but can reduce level of skepticism overall.
3) Well-known HN nick posts "this is my weekend project. I suck at design, but I've written 4 lines of Ruby that can mine bitcoins for you and then mail you an Amazon gift card AND remember to buy you mom/wife/girlfriend a present on their birthday". Good design = These sites never have good design ;) Bad design = This is hard-core real stuff. Just the facts, lightweight simple interface rocks.
Or something like that.
Seriously though, response to design is like response to advertising - subconscious and subtle, something that the user may not understand themselves.
http://tbbuck.com/building-a-web-application-that-makes-500-... was an interesting find, showing a conversion rate jump for a small web app based on positive design.
I think it would be more accurate to measure this in terms of conversion rate or other results-driven measurements.
Personally however, I put a huge amount of stock in a site's design (Reddit excluded).
That's 50% more people sticking around to find out if they like the product.