Combine the beer and brown sugar in a saucepan over medium heat, until the sugar melts and the mixture is slightly thickened; set aside.
Cut the kielbasa into links and place in a slow cooker. Cover with the drained sauerkraut. Pour the beer and brown sugar mixture over top. Cook on high for 4 to 6 hours, or on low for 8 to 10 hours.
Serve on a toasted hot dog bun.