DC Comics' Batman is often called one of the greatest detective characters in fiction, but would he stand a chance of catching Kira? Death Note's Light Yagami is an equally brilliant killer with some supernatural abilities on his side, so it's far from a sure thing. Comparing these two characters and some of their canonical feats and abilities is the only way to find out.
Batman is a character with a long and colorful history, so he has a bit of an advantage in the "feats" department. However, an important part of this question is determining exactly which incarnation of Batman is up against Kira. For the sake of simplicity, it seems using the modern Batman from the comics is the best way to proceed; other eras of Batman may fare a bit differently, however. Batman's abilities include thorough physical and martial arts training, vast wealth and resources at his disposal, and a genius-level intellect that's been reinforced by the best schools. Batman's curiosity has led him to learn as much as he can about a wide array of fields, from forensics to magic to tactics. While this comparison will look only at Batman and Light, rather than their allies and extended teams, the ability to cultivate strong relationships with a wide variety of people is another critical ability of his.
There are two factors that are likely to be most important here, though: Batman's detective skills, and his ability to invoke fear in his enemies. Batman's ability to deduce and investigate leads him to unravel the secret identities of many of his villains, and he's extremely thorough, finding clues that the police or other heroes might miss. Identifying patterns in a series of killings, even if they appear to be of natural causes, is exactly what he's best at. Batman's mastery of fear is a unique tool in his arsenal that could work particularly well against Kira, as fear proved to be one of the most effective ways to provoke Kira into making a mistake.
For Light's part, he's a fairly ordinary high school to college-age student, depending on exactly when in the story of Death Note he'd be from. Light is exceptionally intelligent, although whether he's more intelligent than Batman is difficult to say. Still, he was able to evade capture for years and outsmarted the greatest detective of his own world, L. He also doesn't have access to any special resources aside from the Death Note itself. The Death Note does, however, have a number of abilities that make it extremely difficult to identify its user. It can kill without a trace; there's no way to identify a heart attack of natural causes from one that's the result of the victim's name being written in the Death Note. The Death Note can also kill in an almost infinite number of other ways, including with a delay, which makes it extremely difficult to be sure that all deaths Kira is responsible for have been discovered. Limitations on the Death Note include physical impossibilities (a victim couldn't say, be made to fling themselves into a volcano if no volcano was nearby), the need to know the victim's true name and face, and the need to actually write on its paper. The Shinigami's Eye deal, although Light is unlikely to take it, can make standing against him publicly very dangerous. While Kira primarily kills criminals, that's not an argument that will get very far with the infamously "no killing" Batman. As with Batman, Light will get no assistance from other characters like Misa or Ryuk--something that was essential to getting away as long as he did.
Batman and Kira, in a universe where both existed, would quickly fall into a similar game of cat-and-mouse as occurred with L. Batman would undoubtedly notice a string of criminals dying suddenly of heart attacks, particularly if any members of his rogues gallery were targeted. Batman's secret identity would keep him safe from any immediate reprisals. Light might even have looked up to the Batman prior to engaging with the Death Note, so he may be reluctant to set his sights on what he'd see as a fellow vigilante. That vigilante status may prove to be one of Batman's biggest advantages in such a match-up, as the Kira Task Force had to deal with government pressure and legality (not that it restrained L much). The kind of surveillance Batman can deploy with his money certainly exceeds what L and the task force had access to, which may help accelerate the timetable.
In an ideal scenario for Light, he could pressure government officials to declare Batman a wanted man by threatening to kill them next. Batman would be hunted down, and even if they didn't catch him, it'd cause some major problems for his operation. Batman has one main advantage that L lacked, however: he's already familiar with things like magic and demons and gods, so he'd be able to recognize the supernatural properties of Kira's killings immediately. Kira may have less time to experiment and learn about the limitations of the Death Note as a result. As with L, Batman's identity would be Kira's main goal; if he can uncover the masked hero's face and name, then the game is over. Enough characters in DC's continuity have been able to deduce Batman's identity that it's certainly possible--or even likely--that Light could as well.
Where this gets tricky is with a certain sticking point in the Batman fandom: which identity is the real one? Some say the masked Batman is obviously the alter ego, while others argue that his concept of vengeance and desire to stop crime is so intrinsic to who he is that Bruce Wayne is more of a "character" than Batman is. Even if Light got Bruce Wayne's name and face... it might not actually matter. If Bruce really, truly does see himself as Batman and think of himself that way, then wouldn't "Batman" be the name that needs to be written in the note? Of course, that wouldn't help other people associated with Batman and/or Bruce Wayne, should their identities also be revealed. Kira may well take out many of Batman's friends on his way down, but sooner or later, Batman would find him, and he has the tools to restrain someone such that writing is impossible. If Batman destroyed the Death Note and Light lost his memories of being Kira, he might be in for a very long stay at Arkham. And that leaves one last interesting question: could Light ever be rehabilitated, if he survived a final showdown?
It's hard to say, but Batman would undoubtedly try.