Keep in mind, the Arkham games are all rated "T" for Teen.
That means that the use of "blood and gore" is going to be basically non-existent. Same with language/cursing, which basically is limited to what you could hear on a prime time network show.
It's basically no different than Streets of Rage in that case in terms of level of violence. The only thing you may be concerned about, is obviously the graphics and visuals are light years better....almost life-like. In other words, that realism might make it more "distressing" to a small child, vs. watching obvious cartoony looking character beat each other up.
But other than that, it's fairly tame.