I wonder what the downsides are of putting a copy of one's boarding pass online.
It depends how detailed the boarding pass is!
Probably the most useful information on the boarding pass is your frequent flier number. This is often printed in full. Together with your name, address and other information obtained elsewhere it may be possible for someone to purchase instant flights with the miles on the other side of the planet from your account.
Depending what value you put on your accumulated miles and how many you have, of course this can be a theft running into the thousands of US dollars.
Theft of frequent flier miles is becoming very common and is difficult to prevent, because essentially you rely on the IT skill of your airline. http://www.consumeraffairs.com/news/frequent-flyer-miles-stolen-from-american-and-united-airlines-passengers-011315.html (mirror)
The remaining information could be used to compromise your reservation if not yet flown but there should be no further repercussions without access to other documents; although if combined with other information about you knowledge of your travel history could of course be useful.
Usually, the boarding pass does not contain the ticket number, although the old-style AA boarding passes do. The name and ticket number is sufficient in most cases for someone to access and tamper with your reservation.
Usually the boarding pass does not explicitly (in writing) contain the Passenger Name Record (PNR) locator; a five or six digit alphanumeric code that the airlines use to track your reservation. However, some boarding passes "printed at home" do contain this. This is sufficient, usually with your name, to access and modify the booking, and reveal all other information on the booking. You should treat the PNR locator as though it were a password.
However this information is often encoded in the barcode or in the QR code; so you should blur that out.
If the flight has already transpired; then the risk is only that of information disclosure; of which there is limited use (all they would know is your name, which is easily Googled).
I would blur the reservation number (if its printed on the boarding pass) for flights that are in the future as most airlines require just the last name and booking reference to access complete flight information - which may include your passport details (and passport details of those in your party) if you have supplied such advanced passenger information online.