You just need to build more high-level stuff to raise happiness, so wait till you level up enough to get the necessary perks (Local Leader, Caps Collector, etc.). I used to have quite a few settlements languishing at the 70s happiness for the longest time. But now I'm at level 66 and all my settlements are in the 80s (some in the high 80s), even those that have TVs, radios, and jukeboxes. You just have to build high-level turrets, stores, and beds.
Beds are probably what you build the most in every settlement, so building the best ones will give you a lot of happiness bonus points. That means you need to build houses large enough to accommodate the larger beds.
According to the http://www.amazon.com/Fallout-Dwellers-Survival-Collectors-Edition/dp/0744016312, the stores that give you happiness bonus points are trader stores, food & drink, clothing, and clinics, i.e. weapon and armor stores do not yield happiness (only income).
If you find all the issues of the "Picket Fences" magazines, you will be able to unlock more buildable items such as fancy lights, statues, patio tables and chairs, etc. And these (we assume) give extra happiness as well.
Now the tough part: CHECK IF ALL SETTLERS ARE ACTUALLY ASSIGNED BEDS. Sometimes, even if you have the right number of beds, they still have the "red icons" indicating they are unassigned. You just to have manually assign them, and it may be tough to find the unassigned settler if you have a large population. I've had to do this quite a few times, and each time it DID raise happiness (green arrow showing next to happiness).
Even tougher part: for settlements that have pre-existing beds for pre-existing settlers (e.g. the Abernathy family, the Finch family, the "Workers" in The Slog, etc.), sometimes you notice that those pre-existing beds cannot be moved, stored, or scrapped. And sometimes, they appear unassigned (red icon) too. If they are unassigned, you also need to manually assign them to raise happiness. The even more frustrating part is that sometimes each pre-existing settler HAS TO BE assigned to a PARTICULAR pre-existing bed for it to be successfully assigned (green icon, happiness raised). In The Slog, for instance, Holly has to sleep in one particular pre-existing bed, Jones has to sleep in one particular pre-existing bed, and so forth. If you don't assign them to the correct beds, you don't see the green icon, and you don't raise the happiness.
*Bed placement* also seems to be important. Sleeping bags (the lowest-level beds that supposedly yield the lowest happiness) can be put anywhere since they can be walked on. But for the other types of beds, make sure one side of the bed is completely open for a settler to get on and off it. Don't put them too close to walls. Some walls, you notice, are not entirely flat, and may obstruct a settler getting on and off the bed. Do not put two beds in an "L" shape; put them in "I _" instead. Do not put them in a "T" shape, where one bed is partially blocking another. Put them in "I" formation or the "I _" formation. E.g.:
I I I
I I I
I I I
I _ I _ I
If you have a lot of beds, make sure they can actually be accessed by the settlers: make sure there is a pathway to each and every bed without having to jump (NPCs don't jump).
If beds have placement problem, you may be able to catch a settler standing on a bed or near a bed, which is apparent indication that he or she is unable to get on the bed. And that may affect happiness too.