Unless the property owners agree otherwise, fences on a boundary line belong to both owners when both are using the fence. Both owners are responsible for keeping the fence in good repair, and neither may remove it without the other's permission.
A few states have harsh penalties for refusing to chip in for maintenance after a reasonable request from the other owner. Connecticut, for example, allows one neighbor to go ahead and repair, and then sue the other owner for double the cost.
Of course, it's rare that a landowner needs to resort to a lawsuit. Your first step should be to talk to the neighbor about how to tackle the problem. Your neighbor will probably be delighted that you're taking the initiative to fix a fence that's already an eyesore and might deteriorate into a real danger.