Old registrations had no expiration date, so a pilot could get in a plane that is owned by someone else and verify the airworthiness and registration by inspecting the documents. But that registration might be expired, and without going to the FAA site to verify it is active could find themselves flying an unregistered aircraft. Has the FAA addressed this situation since they started expiring registrations that didn't originally expire? What would the pilot's liability be in this situation?
You verify new (expiring) registrations the same way you verified the old ("for life") registrations -- by looking at the document. You just have to pay attention now (it's not just "It's there", you need to check the expiration date too).
New-style registration certificates (issued some time after October 2010) are supposed to have the expiration date clearly printed on the certificate (In the lower-left corner, under the date of issue).
For certificates without an expiration date printed on them (registered prior to October 2010) the FAA established a table of expiration dates back when the re-registration requirement was initiated.
As regards the pilot's liability, an unregistered aircraft (or one with an expired airworthiness certificate) is de facto unairworthy:
Aside from being a violation of the FARs to fly the aircraft with an expired registration (because it's "not airworthy") this triggers other implications, for example most aircraft insurance is invalid if the aircraft is "not airworthy" .
91.203 and 91.403 make it pretty clear that the owner or operator is primarily responsible for maintaining the aircraft in an airworthy state. If you are flying it, you are the operator and it is up to you to ensure that the plane is registered and airworthy.
You should never get in a plane and fly it without verifying that its maintenance, airworthiness, and registration are up to spec. It's not difficult.
As for checking registration itself, simply check if the document is more than 3 years old. If it is, it is most likely expired.