The other guides posted here give some great tips and guidelines on how to do this. However with most of them you can run into problems with the spotlights seeing you in certain places. This is because you aren't always necessary going to be able to 100% recreate what is shown in the video's. Therefore a lot of the other guides may work sometimes for some people but are not always repeatable on a large scale and could lead to frustration. I believe the following strategy utilizing the secret entrance might be the most consistent and repeatable.
Make sure to grab a Sniper Rifle and some ammo from Ammu-Nation before the mission. Some body armor for latter when you get in base wouldn't hurt either. Make sure you
save before this mission at the airfield (note mission triggers between 20:00 and 06:00). I'd even make
a second backup save in another slot too, just in case something goes wrong. There is no replay mission option and you probably don't want to have to replay the game up to this point.
While your sniping enemies if another enemy starts running around and heads towards you this doesn't count as a spot just make sure to kill him before he gets too close. As long is there is no mention of a code blue/red you are fine. If you do get spotted/trigger the alarm you either kill yourself with explosives or have the enemies kill you to restart the checkpoint. However I personally would just reload my save and quickly restart the mission when this happened to me. With that said when I used the following strategy I completed this first try and even tested it and was able to repeat it multiple times no problem.
The following strategy I recommend is utilized by GamerGuru in this YouTube video linked below.
Just note he states to shoot only the one spotlight in his video but your actually going to have to shoot two. He actually does shoot the two required but just be aware of that and make sure to shoot them both, keeping in mind shooting 3 automatically triggers a code red.
At the start either kill as many guards as you can by sniping through the gate like in the video. Or go crouch through the gate and to the left a bit to avoid the guard in the tower seeing you. Then take out the first guard with a pistol headshot followed by the aforementioned guard in the tower just to your right. Proceed to the left and take out the nearby patrolling guard with the pistol. Followed by all the guards in the towers across from you and the ones in the middle of the map with the sniper. Make sure to shoot out the back left spotlight along with the back middle spotlight. Then continue along the left of the base hugging the wall and killing any enemies left in your path until you reach the middle back tower. Then continue just behind it and up the small hill your right. You will see a bottle on top of 2 grates which you can shoot out with the pistol to access the secret entrance. Drop down here and continue along until the next grate you can shootout. As long as there was no mention of an alarm code blue/red then you should unlock the achievement at the end of the mission. From here continue along the mission but I'd recommend disabling the Anti-Air (SAM site) to make it easier to escape which is optional.