To detect an UFO initially, it needs to enter an inner radar circle. To stay tracked, it must not leave the outer.
Nation happiness has no direct relation to radar coverage. It instead is related to missions. If a mission is spawned outside of radar coverage, it will not be noticed by the player and that nation's happiness will go down at every midnight for as long as the mission is still there (2-3 times). To notice UFO-spawned missions that UFO needs to be tracked by radar while it is shot down or lands. This radar can be a ship - to extend radar coverage somewhat, UFOs leaving the fixed radar coverage can be followed with ships. So, for your base to have an effect, you need to be lucky that the UFO initially flies through your radar and is still tracked when it lands. Or shoot it down while you can, so that it cannot land somewhere where you don't have radar. Sure, often you will be unlucky - if your base is in Europe, missions spawn in America, if it's in America, missions spawn in China and so on.
I'm not sure about ground-based missions, the ones appearing without and UFO - is it enough to cover with outer radar to detect them ?
Due to the above, I'm building my bases in a way that the outer radar coverage is optimized so I don't lose track of UFOs that have been detected once, instead of building them in the middle of nations. This e.g. means putting one onto Hawaii to make the south pacific radar gap small, albeit it covers almost only water.
