My thought, Bacardi1, is that, if my neighbor suddenly fenced me off, I would like to have that neighbor talk to me about what he was doing, and why, just as a matter of common courtesy. I would assume that the neighbor would understand, then, exactly why it was being done. I dunno, I have a great neighbor. We always discuss with each other what we are doing if it will impact, even slightly, the other household. It's not confrontation, it's cooperation.
Talk to your neighbor and explain in detail what you said here and why. Fence in the boundary line with livestock fencing which would keep the dogs out and be suitable fencing if you get horses at home someday. See if your neighbor will spilt the cost and help put it up. Do whatever you have to after the fence is up to run the dogs off, so they know they are not welcome there.
I assume you are separated by acres and not mere feet? In that case, I would just build your yard/play area fence and next time you run into your neighbor outside, let him know how much you appreciate him restraining the dog. It sounds like he is trying and I think that's great. If your houses are close together, like in town, then maybe have a chat to give him a head's up, if you think you might offend him with the fence. But I agree you have to plan for worst-case scenario.
I also agree that a nice looking privacy fence would add property value. Even if it is a smaller area. The next owner may want to use it for their kid, or to let their dog(s) out to the bathroom in. Or, may just want some privacy!
Doing 700 feet of fence that will successfully keep a dog that is motivated, or has time to dig out will be very expensive, imo, in comparison. Start small unless you have a budget that can "do it all" (i.e. dog out for now and good for horses later).
Thank you for your thoughtful replies. Sometimes it is just nice to "talk out loud" about things that come up in our lives and see what other's opinions are. I think we will go ahead and do a tall fence for a smaller back yard area. Privacy fence on that one side and vinyl chain link on the other sides for now. I know that alone will be a hefty amount just doing that.
I will absotlutely mention to the neighbor next time we are within speaking distance how thankful I am that he has indeed been vigilant about restraining the dog and then I will mention that we are going to fence in our backyard to give ourselves some privacy and how it will allay fears of my daughter concerning the dogs. I like the idea of doing a little at a time that one poster suggested too. Perhaps after stage 1, we'll work on stock fence along the borders in a few years, add a shed barn and voila bring the horses home - oy yeah...there is manure pile/removal, hay storage, water, electric, drainage, more fencing, riding ring and on and on and on - can you see why it is easy to get discouraged? Hehehe. Small steps, small steps! Thanks again.