The cause could be many things (listed below). It could be one of them or maybe some combination. The crumbly concrete doesn't sound good, sounds like you have a water problem.
full gutters overflowing and seeping into basement;
broken downspout not directing water away from house
improper grading outside the house causeing water to go toward house and eventually leak in.
the chimney could be the cause. Do you have a masonry chimney? What is the condition of your chimney cap (metal or concrete)? roof and/or sidwall flashing? etc.
Is the window leaking?
If you have never had a moisture problem in the past then something may have changed to cause your current problem. If your neighbor altered their property and is now sending water at your house then he is liable to fix it and restore the previous situation.
You might even have a roof leak that comes down the wall and comes into the basement, weirder things have happened.
Find and fix the water problem and quickly get a decent dehumidifier and pipe the outflow into a sink, sm. sump pump or to the outside. If you try to keep up with the fill-tray instead then you won't be doing it enough - likely. Put on some junky clothes and damp mop every surface in the basement with a weak water/bleach solution if you need to kill the mildew, etc..