John Carter is a beta male (and so is ERB)

   When John Carter first meets Dejah Thoris, he is instantly smitten by her, but is unaware of the social customs of Barsoom and acts awkwardly and ends up disappointing her. Actually, she is kind of disgusted by him.  The second time he sees her, she is struck across the face by a Thark, and this causes Carter to leap into a blind rage and beat the offender to death with his bare hands. Note that he hadn't spoken a word to Dejah Thoris before this feat. He gains possession of her, in an arrangement where she is given without a choice, but she still treats him coldly and indifferently. Essentially, he follows her around as a simp trying to win her favor. Through several more feats of heroism and grand displays of chivalry, she eventually breaks down and warms up to him, and then remains madly devoted to him for the rest of their lives.

    These are not the fantasies of an alpha male, who is at ease around women. These are the fantasies of a beta man, chasing the "one true love" of his life, who believes that grand displays of heroism and chivalry are the keys to a woman's heart. Edgar Rice Burroughs, through John Carter's narration, frequently refers to Dejah Thoris's beauty, and nothing else. He places her on a pedestal, groveling at her image.

    Compare this to Robert Howard's Conan, who has his way with women but communicates with them first, and we get an idea of their motivations and character as a person before any relationship forms. There are also more kinds of women in Howard's stories and not all of them exist for Conan's pleasure, they have their own goals and act of their own accord, and most of them are actually out to trick or use him in some way.

    The whole point of the Dejah Thoris character is to be an unobtainable prize for John Carter. Even though she is physically his slave in the early part of the novel, her emotional distance fulfills that criteria. Once the prize is obtained and she falls in love and marries him, her character has nothing left to do, so all she does in the later novels is get captured and wait to be rescued. This conveniently takes her out of the plot and restores her unobtainability, giving Carter back a prize to chase until the end of the book.

    In A Princess of Mars, we are told that Barsoomian women are always armed with at least a dagger, as life in Barsoom is violent and dangerous. We are also told that Dejah Thoris is a princess, leader, and was in command of a scientific expedition. The image of Dejah Thoris is completely at odds with what she actually does in the novel. I don't think she even fights Phaidor, but only manages to survive a catfight.

  John Carter is not a fully formed character - he is a self insert for the audience. Nobody reads the novel for John Carter. Burroughs knew this and that's why the first novel is titled A Princess of Mars. He is the closest to the ideal of a chivalric knight that was contemporary to Burroughs's society - a Confederate Cavalry Officer.

    The audience wants Dejah Thoris to be a kickass warrior woman from a far off planet, but she has no space to be so in the context of the story. This is actually an issue for the later novels. Once John Carter has established himself as the Warlord of Mars and taken Dejah Thoris as his wife, what is left for either of them to do?

   The later novels introduced the adventures of John Carter's children, but none of them are as exciting to read about. They also do not have the colonial fantasy that the original Barsoom trilogy does. This is evidenced by all modern revivals, such as the Marvel Comics run from the 70's or the current Dynamite Comics ones, which return to creating more adventures for John Carter and Dejah Thoris to be in. In Dynamite Comics' case, John Carter is ancilliary to the story. Several of their miniseries have him disappear completely and the focus is placed squarely on the adventures of not-quite-single Dejah Thoris.  Unfortunately, it seems like their writers have no idea what to do with her, either.

   I can suggest two methods of driving interest back into the character. The first, cheap method, is to just do what shonen anime does, and turn Dejah Thoris into a fanservice character that hangs around the main character while he gets into fights and does cool stuff. If you look at the covers to the comics, you'd be forgiven for thinking that this isn't already the case. All the covers are just pinups of Dejah Thoris.

    The second, better method is to give the audience what they want, and revert the emotional dynamic of Carter and Dejah Thoris's relationship to what it was in the beginning of the original novel. Make Dejah Thoris the unobtainable warrior queen that the fans want her to be. As a royal scion of Barsoom, she will always be the thing that Carter could never have, and as an accomplished warrior herself, mere acts of bravado are not enough to impress her.

   Even if the stories take place later in the timeline where the two of them are married, Carter will remain in awe of her, and will constantly be trying to prove himself to her. This will actually return Carter to being relatable to the audience. Nobody relates to the warlord of an entire planet.  It also gives Dejah Thoris the freedom to act in her own stories. As leader and warrior, she can have her own adventures.

To make Barsoom stories interesting again, make Dejah Thoris an alpha female.

I think the fan perception that John Carter is a manly man amongst men, and Dejah Thoris is an empowered woman, is completely false. I wish someone would write that story, though. It sounds kickass.