How close is the Geneva Airport train station to the airport?

The train station is at the airport. You have to take up two (or three) escalators from the platform and then walk for approximately 50 meters to arrive at the check-in. There is only one terminal, so it will take you 5-10 minutes at most to go from the platform to check-in. It is a rather small airport and you should be able to find it without problems.

This being said, the airport says that you have to check-in 40 minutes before the flight leaves and be at the gate 30 minutes before departure. I am not sure how much that is enforced. I'd be wary especially if you are flying Easyjet, as their gates tend to be very far away. Also note that there can be quite long queues at security.

I would say you can definitely make it to the check-in in time, what comes after will depend on how busy the airport is and will require that you don't have any problems at immigration or security.

Train station is actually next door. The walk takes 5 minutes from platform to check-in. Here are some tips to save time:

Board as close as possible of the back of the train related to the the direction of travel; the airport station is a dead-end and the exit towards the terminal is near the tracks "exits", not the buffers. From Zürich HB, train should keep the same direction of travel whether using the Bern (IC1) or Olten/Soloturn/Neuchâtel (IC5) route. Just saying that because some trains are reversing halfway on some routes.

When you are in the train station hall (a kind of mall with multiple shops), do not ride the second escalator up but head to arrivals levels instead. From the arrivals hall, then, go up to departures. This will save you the need to go outside and wait to cross a busy road.

Leaving from French sector (gates F, mainland France destinations)? Allow significant extra time if you may depart during rush hour! French sector is accessed by a customs checkpoint on landside (before security). There are a few check-ins counters there which are not too busy but the struggle is the security itself. With 2 open lanes in average, I already had to wait in line for 1 hour on a busy Friday night to catch an easyJet flight. Would have missed it if it was not delayed. And the area past security is damn small in that sector; there is only one small shop and one small food take-out place.

Main (Swiss) sector has many more available security lanes and extensive shopping/dining options.

It's really in the airport (which is itself not very big but not so efficient for its size). Arriving: Out of the baggage claim, take left, walk 50-100m tops, you are already in the station (the actual tracks run underneath the airport). In the other direction you need to take some escalators, find the right check-in desk and then take some other escalators to reach the security check but it's not very long either. You will walk more between the security screening and your gate.