The style of The Beast in the Jungle is realistic yet ornate. It contains lots of long, flowing sentences with many commas. It also tends to emphasize characters' psychological states. By placing great importance on the psyche of a troubled man named John Marcher, James eschews the romantic elements of life in favor of a real struggle that someone might face in real life: the tug-of-war between passivity and action, as well as between the decision to wait for one's fate rather than taking it into one's own hands.
It is also important to note that much of the dialogue feels a bit stiff or artificial, but this may just be a feature of 20th-century British speech or a feature of James' writing, as his characters often speak in a similar manner as May and Marcher. Their dialogue often has a rather blocky, choppy, awkward structure that reflects John's hesitancy and May's reticence. It also contains more complex sentence structures that appeared more in early-20th-century English speech than they do in the speech of people today. Despite the fact that people in the present day don't usually speak this way, one could imagine people doing so in Henry James's time.
