Deep Space Nine is darker and contains higher levels of emotional intensity and drama than other Star Trek shows. But it never gets to the point of severe. It's still very lighthearted when compared to scarier shows.
The story peripherially involves: racial opression, slavery, genocide, internment and concentration camps, ghettos (deceitfully called "sanctuaries") with humans in it, secret agents and spies, interrogations, organised crime, suicide, terrorism, weapons of mass destruction, possession by supernatural forces (both good and bad), mental illness, demon worshipping, suicidal cult, coup d'etait attempted by a human military officer, treason and theft committed by a number of defecting human military officers.
One torture scene (see in violence section). The background music extends the psychological effect.
The series feature a species of malevolent shape-shifters who have the ability to undistinguishably duplicate people and objects. Might be frightening for younger children.
A lot of alien species appear in the series, some of them (like the Letheans or the Jem'Hadar) may be frightening for younger children.
The episodes " ...Nor the Battle to the Strong" (5x04) and "The Siege of AR-558" (7x08) are "in the trenches" kind of war stories that may be too intense/realistic/dark for younger children. The latter involves a boy permanently losing a leg as result of a war wound.
The episode "Distant Voices" (3x18) may be particulary scary for younger children.
The episode "Hard Time" (4x19) contains high emotional tension, involves confinement, starvation, murder, family/social disfunctionality and suicidal intent.
There are a few other episodes - like "The Visitor" (4x03), "The Begotten" (5x12), "Dr. Bashir, I Presume?" (5x16) or "Time's Orphan" (6x24) - which are emotionally intense.