Joe is the narrator's boyfriend. When we first meet him, he seems a bit impatient and not necessarily super-interested in or aware of his girlfriend's emotional state. For example, when the narrator rejoins the group after going to check in with Paul regarding the "missing dad" sitch', she reads his greeting of "Any news?" as a "neutral mumble that signals he'd prefer if I kept from showing any reaction, no matter what has happened" (3.16).
Of course, we don't know how much we can trust the narrator, since she proves herself unreliable later (and if she were really a good judge of character, she probably wouldn't be traveling with any of these people in the first place—or dating someone she thought was that callous). But since her narrative is what we have to go on, we have to at least float the possibility that Joe isn't exactly Mr. Sensitive.
The narrator seems to like Joe best when he's not talking, celebrating his furry, heavy body a lot more than his personality: "Everything I value about him seems to be physical: the rest is either unknown, disagreeable, or ridiculous. I don't care much for his temperament, which alternates between surliness and gloom, or for the overgrown pots he throws so skillfully on the wheel and then mutilates" (6.24).
Yes, you read that correctly. Apparently Joe likes to do violence (artistic violence, anyway) to pots, turning them into these wild, hole-ridden monstrosities that the narrator can't even put flowers in because the water runs out of them. By the way, Joe teaches his perhaps dubious skills in ceramics (and poetry—really, Mr. Sensitive teaches that?) in the same adult education program where David works.
Joe's kind of a brute, and it shows in his sexual behavior. He comes close to forcing himself on the narrator at one point when she's telling him she doesn't want to get frisky:
"Don't," I said, he was lowering himself down on me, "I don't want you to."
"What's wrong with you?" he said, angry; then he was pinning me, hands and manacles, teeth against my lips, censoring me, he was shoving against me, his body insistent as one side of an argument.
I slid my arm between us, against his throat, windpipe, and pried his head away. (17.36-38)
The narrator doesn't really make a big deal of the incident, but refusing to stop when his girlfriend said "No" and pinning her down is not kosher behavior. Then, when she "pries" him off, he goes off in a huff. And then sleeps with Anna soon after—because that's what you do when your girlfriend says no to sex a couple of times? Very uncool.
With that all said, over the course of the story, the narrator seems to decide that he's a good guy. In fact, in the book's final lines, she pays him her highest compliment by saying that he's not an "American" (27.9) and even admits that she loves him, which was a huge sticking point for her earlier:
I watch him, my love for him useless as a third eye or a possibility. If I go with him we will have to talk, wooden houses are obsolete, we can no longer live in spurious peace by avoiding each other, the way it was before, we will have to begin. For us it's necessary, the intercession of words; and we will probably fail, sooner or later, more or less painfully. That's normal, it's the way it happens now and I don't know whether it's worth it or even if I can depend on him, he may have been sent as a trick. But he isn't an American, I can see that now; he isn't anything, he is only half-formed, and for that reason I can trust him. (27.9)
We're not sure if "half-formed" and "he isn't anything" count as compliments in our universe, but the narrator definitely intends them as such; even though she's decided to turn to the modern world, it seems she might still appreciate people who haven't quite "evolved" fully from their animal state—like brutish, animal-like Joe.