Call the city. If the water is stagnates it can cause issues. There is most likely a regulation that is being broken.
This pipe might be the water companies responsibility and they are not wanting to fix it. The last water line repair in our neighbor due to leaky water line due to old pipes 15 homes got new main lines. Our city is watching foreclosed homes carefully. What is happening the water gets shut off this can break the old lines causing leaks that go unseen. Or when they get cut back on it causes leaks. Our city has to bully the water company a few times to fix damage and not over charge property owners. The water company argued for 2 years about who was responsible for a water line in a park (that has no water service - it just runs through the property).
Or another possibility is your neighbor's inside house is so messed up he cannot have people in to properly maintain (watch an episode of Hoarders). But since it is outside the house the water company might be responsible.
Another thought...........are you sure it is water not sewage? Again this can get into a "who pays" "who is responsible" These can be expensive repairs. Again, our city was proactive and we pay extra a month for sewage however, outside repairs due to old pipes are covered.
It sounds like you haven't addressed this directly with the neighbor. Believe me, I am the LAST person to step easily into these kinds of confrontations, so I understand why you might want to avoid the discussion. But you really have to ask them what the deal is. You can ask in a non-confrontational manner. I would think all that wetness is causing / will cause their basement to be damp and moldy, not to mention all the money wasted. So you can also frame it in a way that looks like "Wow, aren't you worried about mold?" type thing. You can also just say it's unsafe for your kids to be rolling over it and you want to call the city / water company and find out if they are responsible (where I live the city would be) and you need to first know what they have already done / found out. Then start pursuing it with the city yourself. If for some reason they are not responsible, then you can go from there how to deal with your neighbor.
Mama since 2010
Multicultural living in Europe
I would think the sewage drain is the more likely culprit, too. I'd think water going into the house would just peacefully drain down into the earth and you'd never see anything. A blockage in the sewage pipe will cause bursts of overflow up onto the sidewalk. And if it is sewage, you really have ammo to call the township/city and have them bring pressure on your neighbor to fix the problem. A couple houses around here have had problems with that. The first one took months and months to get fixed, maybe a year. You'd see paper bits (dissolved TP) on the sidewalk and street near the little pipe vent cover. It was across the street so we'd just walk around it.
Here's my local page and leaky, smelly things are code violations.
Call the city. I know for our area you can even do a complaint such as yours online.It falls under health issues and sanitation. Your neighbor might get a ticket or repair it,or the city might have to repair it.Either way someone will fix it.Just act clueless so your neighbor will not suspect you as you do not want the wrath of a nfh.