Although the portrayal of Cheng in Broken Blossoms can be seen as positive, it carries assumptions about racial identity and gender that are in fact very negative; the film ultimately suggests that Asians can't cope with modernity and that their ways are backward.
Cheng is portrayed as an effeminate man, thus equating Asianness with femininity. He looks frail and soft, in stark contrast to the rough manliness of Battling Burrows. He always stands in a way that takes up minimal space, and never asserts himself in his surroundings by taking up space--an attribute associated with masculinity.
Furthermore, he walks hunched over and with a limp--a portrayal of the physical weakness of the Asian. His frailness has more in common with Lucy than with any of the other men, and they share a bond because of this element of physical weakness.
Cheng's mission is to spread Buddhism to the West, but the treatment of his journey carries imperialistic and conquistadorian overtones. While one could read the portrayal of Buddhism as peaceful and positive, the film in actuality seems to suggest that Buddhism is idealistic and backward. The modern world, represented by the West, is rough and harsh. Once Cheng encounters the "real world" of the modern West, he's forced to take violence--his ultimate shooting of Battler represents not only his defeat of this enemy, but also the defeat of his ideals. His peaceful Buddhist ways prove to be unworkable once he leaves the Edenic East, and ultimately he loses the battle to keep his own values alive.
His missionary role is ironic, since in the end he is the one who becomes converted. I saw a parallel between this process and the colonization of the New World. Europeans, upon discovering Native Americans, found their peaceful, communitarian, spiritual ways to be backward and untenable for the modern world as defined by Europe. In the same way, Cheng finds that his own ways of peace are unworkable. If the blossom represents idyllic peace, Cheng himself is the broken blossom--his old conception of the world is shattered, but unfortunately he is a weak, effeminate Asian man who can't really make it in the real, modern world. Thus, even though he shoots Battler, he's left with a dead girl and a broken spirit. His own realization of his condition causes his suicide.