The problem with the Broadcom chipset isn't the kernel and a newer kernel won't fix it. When SalixOS 13.37 was released the Broadcom firmware was proprietary and their license made it impossible for the SalixOS devs to package it. It is now permissible to package the firmware as recent versions of Fedora, Gentoo and Ubuntu have done. I imagine it will be included in the next release of Slackware which means it will also be available in the next release of SalixOS.
In the meanwhile you can use slapt-src or Sourcery to install it using the script at http://slackbuilds.org/repository/13.37/network/b43-firmware/
. You'll also need to install b43-fwcutter which, if memory serves, is already in the repository. The netbook I'm using right now works perfectly well with SalixOS/Slackware 13.37 with the firmware added and it has a Broadcom 4312 wireless chipset.