IANALY, either, but the OP isn’t so much concerned about potential liability to guests for injuries caused by the barbed wire, but rather simply wants the barbed wire that trails off onto his property removed. That is, can a landowner compel a neighbor to remove part of a fence that overhangs his land.
The common law rule in this situation was cujus est solum, ejus est usque ad coelum et ad inferos; the idea being that land extends infinitely upwards and downwards. Hence, any kind of overhang from a structure on adjacent land would be an ouster of the landowner’s property rights, and thus constitutes a trespass. At common law, a landowner could then compel the removal of the encroaching structure; in this case, the fence. In more extreme cases, the common law courts would require entire houses to be torn down because they extended six inches across the property line.
As we might well imagine, the common law rule does not work well in the modern world. The idea that one’s airspace extends to infinity is incompatible with satellites, airplanes, etc. Nevertheless, the OP could still probably maintain an action for trespass, though this will vary depending on the jurisdiction.
It is the remedy for trespass with which the OP is concerned, however. Can the OP force his neighbor to remove his fence? At common law, that would be the normal remedy. Modern courts, however, sometimes exercise their equitable powers to shape an altogether different remedy: a forced judicial sale of the encroached-on property to the neighbor. Courts are not eager to contribute to economic waste.
From the description of the facts the OP provides, however, it does not sound like the courts would favor a judicial sale. Removing the encroaching parts of the barbed wire fence would not constitute economic waste – assuming, of course, that the OP is correct on where the property line is drawn. Moreover, a shoddily-constructed barbed wire fence can pose a danger to the OP and those on his land. The equities here seem to be in favor of requiring the removal of the offending portion of the fence.
An altogether different question is whether the OP can resort to self-help. That is, can the OP just remove the trailing barbed wire himself without asking permission from the neighbor? Unfortunately, I have no idea. Assuming that it could be done without damaging the fence, I don’t see why not. However, if one damages the fence, liability for the cost of repairs would not be out of the question. Moreover – not to impugn the OP’s skill at barbed-wire-fence-pruning – there’s also a distinct possibility that one could hurt oneself in the process. By having the neighbor remove the offending portion of the fence, at least the OP precludes the possibility of being injured in the process.
Ultimately, though, the goal is to have the neighbor remove the overhanging portion of the fence without resort to the legal system. Lawyers and courts antagonize; more gentle forms of persuasion may be in order. As the others have suggested, asking the neighbor to remove the portion of the fence is the best solution. The neighbor will probably be more willing to agree if the request is couched in terms of safety to pets and children on the OP’s land rather than in terms of property rights.
What is it that Robert Frost said about mending walls and good neighbors?
Best of luck.