I am evidently way more concerned about property lines than most of the neighbors I've had. DrBoost...I assume the same is true for you. (This doesn't mean we're wrong).
As mentioned, adverse possession is a real thing. I wouldn't want to have to go through that process and the cheapest, easiest, fastest way to ensure you don't have to is to make darn sure the fence is built in the right spot.
It sounds like you've already satisfied yourself that it will be built on the property, so that's good.
Now about the cost sharing. If they don't talk to you about the cost before it's built, then all bets are off. Of course it is neighborly to help pay for the fence (IF it's a fence you want...and in this case you clearly do value having a fence there). If they come to you afterward and ask if you'll share the cost one way to approach it might be:
"I got a quote for $XXX to build a fence separating our properties. I shopped around to ensure this is the most competitive price for good quality work. I'd be happy to pay half of that amount if that sounds ok with you. What are your thoughts?"
Of course, it would be even better to have this conversation BEFORE the fence gets built. If they're not willing to initiate the conversation you might* want to. It will be slightly less awkward beforehand than after, in my opinion.
*I'd say it's entirely possible that they don't want to have the conversation and have no intention of ever asking you for a contribution. And if you're comfortable it's in the right spot and aren't too concerned with the expected shoddy craftsmanship, that's a bet I'd probably take, lol.
I've only built a couple of (residential) fences myself and I've never asked anybody to help pay for them. To do so, in my mind, would be a bit presumptuous about their priorities.
Story time:
I had a neighbor start to build a fence not even remotely on the property line. I had to stop them and make them do it right. The funny thing is that THEY had PAID for a survey that clearly showed were the property line was. When confronted about it they said they were told (they probably weren't) by their realtor where the property line was (incorrectly). Having been in the real estate game for some time I had the resources at my fingertips to show them where the property line was. It helped that I had a pretty darn good idea already where the lines were based on nearby pins in the neighborhood. It REALLY helped that they had recently had a survey done on their own lot. I simply walked them around their lot and pointed out the clearly marked corners.
Another neighbor, just last year, was super nice and REALLY wanted me to give him the go ahead to build "a really nice fence" (vinyl...barf) between our houses while fully admitting he had zero clue where the property line was. I smiled and insisted that we needed to know where the property line was for sure before a fence was built. He still really wanted me to give him the nod to "just build it where it seemed natural" between our houses. Again I smiled and said that we really needed to find the proper property line. I offered to help with that (again, I had a pretty good idea because evidently this is WAY more important to me than most folks). In the end, I got out my metal detector and found the survey markers for him and he was very enthusiastic about that...and built the fence in the right place.
These are just a couple of examples...I could go on and on.