Mold is problem for modern construction techniques. Mold is ubiqutous in the UK, but wasn't a problem for buildings themselves until the advent of contemporary (English) construction techniques. Those hundreds-of-year-old homes built of stone don't have issues with mold (their inhabitants might). It doesn't take much more than ensuring proper airflow to mitigate mold issues in actually temperate climates.

Now, mold in, say, the SE USA (or even the mid-Atlantic region), is a different story, but once again, well built homes there from the 1800s can and have avoided mold issues with good air flow design.