My bike's adjustable stem keeps falling down
What you have is an adjustable stem. The headset is the pair of bearings in the head tube that the fork steerer tube runs in, that allows the handlebars and fork to turn.

You are tightening the correct bolt, however, check that the nut on the other side is captive and not turning when you tighten the bolt. If it is not captive you need to hold the nut with another wrench as you tighten the bolt.  

These types of adjustable stems usually have a a series of radial splines on the inner and outer parts that lock into each other when the bolt is done up. The splines - being made of relatively soft aluminum alloy - get worn down and deformed, and can no longer hold the forward part of the stem in place. If yours is repeatedly slipping the splines are probably very worn.

The solution is to buy a new stem. You can either find a replacement adjustable one, or knowing where you like the bars to be get a fixed one that replicates that position.

Replacing the stem is relatively easy and videos on how to do it are easily found. The tricky bit is preloading the headset bearings. You have to get this right or the bearings will be damaged when you ride, or in the worst case the bike will be dangerous to ride.

You also need to replace the steerer tube top cap. This is necessary to set the bearing preload. Even if you don't replace the stem the lack of top cap is allow water to enter the top of the steerer tube and accelerate corrosion.