Only 12 minutes to change Trains in Brig Switzerland.

Swiss trains run on time. Twelve minutes will be plenty. You can download a plan of the station from this webpage: https://www.sbb.ch/en/station-services/railway-stations/brig-station.html

There may be a way to find out the usual platforms used by your trains, but I don't know how to do it. In any case, I'd still check the electronic sign on my way to the new platform to be sure everything is as expected.

Here are some helpful tips for making successful train transfers in minimal time.
1) Everyone handles his/her own luggage and can carry it on and off the train on his/her own.
2) A few minutes before arrival, make your way to the end of the train car and be ready to exit upon arrival.

Where are you coming from and where are you going to? www.sbb.ch/en will tell you what tracks you change to. Some times, you just get off the train, and as soon as it departs, your new connecting trains pulls up. That is optimistic, but the SBB site will tell you. Also, if you are coming from/to Zermatt or Andermatt, the trains stop in the street outside the station and you have to go in or out of the station for mainline trains.

Some German trains give you only 4 minutes to get off a train, run down a steep flight of steps, march across the train terminal, rush up a steep flight of steps, and jump onto your next connection. When a train schedule says it departs at 12:00pm, it means 12:00pm and ONE second, not 12:00pm and 59 seconds.
The ticket selling machines can print your schedules, in English, which states which track you will arrive on, and from which track your next train departs.
Just be prepared and dash. I assume there are elevators, but in my experience, they are harder to find.

12 minutes is an eternity in changing Swiss trains. Your train from Geneva is probably terminating at Brig, and is designed to connect with the train to Milan (coming from Bern).
As the others say, you can look up platform numbers on the SBB site here: https://www.sbb.ch/en
And the station plan is here: https://www.sbb.ch/content/dam/infrastruktur/trafimage/bahnhofplaene/plan-brig-a4.pdf

"I assume there are elevators, but in my experience, they are harder to find."

Elevators are only present at some stations - never assume a station has them without specifically checking.

Looking at the map Chris posted of Brig station, it seems they have ramps as well as stairs, but I don't see elevators. So, if you have difficulty with stairs with your luggage, seek out the ramps. If your next train is not across the platform, you will go down to the corridor connecting the platforms, through the corridor to your new platform, and then up to the new platform. There is no security or ticket checking, so these aren't present to slow you down.

Do be ready to get off BEFORE your arrival. Otherwise, you'll be "swimming upstream" as you try to get off when others have started getting on. Unlike on a plane, you do not wait until the train has come to a complete stop before gathering your belongings and heading to the exit.

Unless you have a mobility impairment, you should be fine with 12 minutes (from the map, Brig isn't that big a station).

The ramps at Brig are relatively steep but well lit and well signed. As long as you don't wander down into the shops area (one bakery - very tasty - for both take away and sitting down at a table) you can change trains multiple times there in the time you have. 4 or 5 minutes would be plenty.

For other readers, if changing to or from the Glacier Express, as mentioned above, I'd want a quarter of an hour because it does involve going outside to the middle of the road, and down a bit.

But for all mainline trains it is an easy connection at Brig.

As with all Swiss stations pay attention to letter as well as the platform number - i.e. 4C. Platforms are divided into as much as A-B-C-D and the bit of the train you need, or sometimes the entire train, will be at a particular letter. This is particularly important at larger through stations such as Basel SBB, Bern, Spiez, etc. where there may be a train behind another so you can get on the wrong one if not careful.

Enough time for a nap in the middle.

Are you talking about the 12:12 departure from Geneva, connecting to the EC 57 at Brig at 14:32-14:44? You should arrive on Track 1 and the EC 57 will go from another track, like 7 or 9, depending on the day. So when you arrive at Brig, note the connecting track on the sign board on the platform and go down the ramp/stairway to the crossover tunnel, walk maybe 50 or 100 feet to the stairway/ramp for the new platform then up to the new track. That should take about 3 minutes. EC 57 stops for 4 minutes at the station, so you should see it arriving as you get there. They may also announce the connecting track on the train PA just before you arrive. And you can also ask the conductor.

You do know that there is a direct EC 39 departing Geneva at 13:39, so you would not have to change trains at all.

We changed trains there in less than 9 minutes yesterday..

Consider yourself lucky with 12 minutes in Switzerland! Try the 4 minute change from grutchalp gondola to lauterbrunnen train...that one is a killer!

As long as it’s not an Italian train you should be right. Five days ago we were going from Domodossola in Italy to Brig on an Italian train and it was about half an hour late so we missed our connection in Brig.

Then yesterday we were waiting for a train at Thun and saw an Italian train pull in about 20 minutes late. There was an announcement that the connecting train couldn’t wait and all those people expecting to connect with it were out of luck.

There was an announcement that the connecting train couldn’t wait and all those people expecting to connect with it were out of luck.

Not totally out of luck. All rail lines in Switzerland have at least one train per hour, many one every 30 minutes. So the next train would be less than 40 minutes wait.
The Swiss and Italian railways have a very different understanding of what "on time" is. Often, if an incoming Italian (or French or German) train is very late or cancelled, they put on an extra train to fill that timetable slot inside Switzerland.