At Standard FNM, Adam plays swords to plowshares on Nina's creature.
Nina calls a judge, saying thats not a legal standard card.
Adam shows the Judge on Gatherer that it is indeed standard legal, citing Wizards as the source: (as of time of this posting)
http://gatherer.wizards.com/Pages/Card/Printings.aspx?multiverseid=382380

What is the correct way to handle it?

As noted elsewhere (http://apps.magicjudges.org/forum/post/65930/), this is currently an issue with Gatherer; it is being addressed, as quickly as possible.

It remains a player's responsibility to understand deck construction rules, as stated in the Magic Tournament Rules; Conspiracy is not one of the sets listed as legal for Standard (nor for Modern), and as such, cards from that set can only be played if they were also printed in one of the Standard-legal sets.

In your example, FNM is a Regular REL event, so you help the player correct his or her deck so that it becomes legal for play (per the Judging at Regular REL policy guide).