Romantically partnered up with Alice, Jasper is the friendliest Cullen brother to Bella. From the baseball game onward, Jasper is Alice's accomplice in keeping Bella safe while James is tracking her. His presence is helpful, since his sixth sense allows him to feel (and to some extent manipulate) others' emotions. This talent makes him an empathetic and calming presence to Bella, who's not exactly happy or calm through the adventure to Phoenix in Chapters 20 through 21. (Though Bella does get sick of feeling like he's altering her emotions.)
It's important to note that while Jasper is decidedly friendlier than Emmett and Rosalie, he isn't as close with Bella as Alice is, by a long shot. It seems he still has issues with self-control around Bella. For instance, when Alice says that she and Jasper will take care of Bella, Edward questions whether Jasper can "handle this," meaning, handle being around a human. Alice responds, "Give him some credit, Edward. He's been doing very, very well, all things considered" (19.165, 166). All the same, Jasper and Emmett both have to flee the dance studio where Bella is hurt because there's so much blood. Jasper is interesting in that he actually evolved into a "vegetarian" vampire after an unspecified amount of time as a human-eater – somehow he developed a conscience, just like Alice. After that "adaptation," he moved in with the Cullens.