I think there are threefour reasons why dwellers get unhappy:
- Injury: If they are injured, heal them with stimpacks
- Hungry/Thirsty: Make sure you have enough water and food (the indicators should not be red)
- Job: Make sure they are assigned to the correct job. Check the SPECIAL attributes and place every Dweller in a job where he/she has the most points at.
- Irradiated: If they are irradiated (they have a red bar on their health meter), remove radiation with RadAway.
If you zoom in on a particular Dweller, they will talk, and say why they are unhappy.
One of the tips in the loading screens said a Radio Station makes Dwellers happy. Not sure this increases the general happiness or only counts for those Dwellers assigned to the Radio Station. There might be other structures which will increase happiness. I have seen there is a garden (but I can't build it yet to confirm).