This tends to lead to mold, rotting floor joists/sill plates, and other moisture based damages and problems.

I would love some cheaper design ideas for the basement, but I want something that will stand up to moisture issues.

I assume accommodations got a little crowded :)Two problems:1) Water bill goes out of sight2) Lot of extra moisture from too many people in the unit means more likely to get mold. 

I have used DE inside for flea and tick treatment with some success but keeping your yard tick free in an area with grass and high moisture probably not going to happen.

Are you certain there's no new potential causes of moisture in the home?

Not sure if it our dirt, or the moisture in the soil, or maybe the jalapeños and hot sauce we eat.   

I also add a automatic bath fan in all bathrooms to get that moisture out.

Someplace that old needs a thorough inspection of the foundation,roof,electrical panel and wiring,plumbing,water heater,and HVAC.Especially in a harsh weather climate like Wisconsin or anywhere up north.You really wanna check for long term water damage to the foundation and base of the property caused by years of snow and ice, no moisture barrier installed,and poor drainage.The cost of repairing these items should be factored into your bid price.You didn't say what the after repair value of the property was,only that they were asking 135k.If the ARV is only around 135k after improvement costs then 100k and just cosmetic changes would be decent.However,if you find after a thorough inspection that the place needs far more work to get up to code and up to stuff,then you need to show the owner all the problems and negotiate the cost down to cover those repairs.

Duplex was completely remodeled but one of the walls started to have some "moisture patches"It is only a portion of 1 wall and only in the downstairs unit.

There are multiple ways to handle moisture from exterior preventative systems to interior French drains.