If the fence is yours on your land and the deeds state the fence is your responsibility you have to maintain it.
However your neighbour cannot attach anything to it, paint it or damage it in any way (without your permission).
Do they realise they are recking your fence? and are they doing it on purpose?
If they are simply unaware of the law and the damage they are causing then you could, in the first instance, go and ask them not to do it and explain why.
I'm not sure about asking for payment is enforceable but you could try. Mind you if after being told of the damage they are causing they carry on, then that might be seen as criminal damage?
Unless stated in the deeds of your property that you HAVE to maintain a fence then no one can make anyone else do put one up or keep in in good order.