Most axle tubes, especially older ones, are readily welded the same as you would for structural steel. I would vee out the crack and go a little beyond to be sure you got it all. I would degrease the area remembering that grease will run into all kinds of places once it gets hot from the weld- which means it will run into your weld if you don't clean back far enough. I would also heat up the area to burn off anything you didn't get in degreasing. If you can reach inside, I would put ceramic backing on the root side. My preference for filler metal would be E7018 simply because the weld metal often has better ductile properties that many MIG wires will give you. One final, thought is to stiffen the box tube by clamping on an angle or such, possibly even tack welding a flat bar on edge on the opposite side. This is to reduce distortion you will get from welding.
CHGuilford

I can't add anything to what GHGuilford has stated. I agree. What I can add is that the local axle manufacturer welds these axle housings with 7018 when any repairs need to be made.