Star Trek - Khan and Chekov

Space Seed was supposed to be a bottle episode. As few of the regular cast was to be used as possible. Since the Enterprise was on a 5 year journey, we would presume the ship didn't run back to HQ and drop everyone else off; rather only those few regulars were shown as being instrumental to that particular script.

In this case, only Kirk, McCoy, Spock, and Scotty were shown. Uhura, Chekov and "Oh My!" err, I mean Sulu were not. This was to save money for the hiring of the Guest Star, Ricardo Montalban, who portrayed Khan.

In universe, this would not mean that all of the other crew wouldn't know what was happening, rather they just weren't instrumental. Chekov would still have known about Khan and his crew, as would Uhura and the others not shown in the episode.

Not to compete with the above answer by CGCampbell, but just to add some further information, Memory Alpha has the following to say on the matter:

Khan's apparent recognition of Chekov and his remark "I never forget a face" are somewhat ironic, since Khan's appearance in TOS: "Space Seed" was in the first season and Chekov did not make his first appearance until Star Trek's second season. It is possible, however, that Chekov was on the Enterprise at the time and Khan had seen him off screen. In fact, as noted in the Special Edition DVD's text commentary, Walter Koenig often joked (at conventions and in interviews) that his character had made Khan wait overly long to use a bathroom on Khan's visit to the Enterprise and that was why Khan remembered his face so well.

This really doesn't present a problem at all. The Enterprise had a crew of over 400 people, the vast majority of whom we never see. Chekov was obviously a crew member, just not one of those precious few we get to see until his more prominent assignment (on the bridge) in seasons 2 & 3.

Also why is Chekov in "Star Trek Into Darkness"?

Star Trek into Darkness is in another timeline refered often as the Kelvinverse due to "Kelvin" (in this case the U.S.S. Kelvin) being a recurring name reference in Bad Robot works with Kelvin being a reference to JJ Abrams grandfather of the same name...

SPOCK: You are assuming that Nero knows how events are predicted to unfold. To the contrary, Nero's very presence has altered the flow of history, beginning with the attack on the USS Kelvin, culminating in the events of today, thereby creating an entire new chain of incidents that cannot be anticipated by either party.

UHURA: An alternate reality?

SPOCK: Precisely. Whatever our lives might have been, if the time continuum was disrupted, our destinies have changed.

The timeline was first created in 2009 Star Trek (the previous film) where in the future of the prime timeline, Ambassidor Spock fails Romulan from being destroyed by a nebula. A survivor with his crew named, Nero seeking revenge, uses "red matter" to create an articifical black hole and they and Ambassidor Spock time travel back in the time line (at different times), creating an alternate time line (possibly a branch off. Most Bad Robot shows that us time travel opperate this way and I am pretty sure Alex Kurtzman has been quoted saying the new timeline runs parallel to the prime, suggesting it is not "gone").

With that, the point (much like Fringe and possibly LOST) is that a new chain of events ensue and challanges what can change and what can stay the same looking at the alternate TOS characters. It's basically a look at identity through free will vs fate (or reciprcoity), when one has to consider a timeline being made from another...

So Chekov can be there, because the events of how the TOS characters come together are not the exact same.