I live in dry Colorado so basement moisture isn't my area of expertise - but here are a few things to consider.

You can also see that the dark spots on the tiles are moisture (it's been raining hard here.)

So far, no problem with moisture or other (although scuffs do show more on the whitish Coventry Oak, also ordered online.

It's a cooler/cold climate (western Washington state).I upgraded the bath fan to 100 cfm (double previous fan's cfm) and added a timer switch so that they could leave the fan on to go to work/school.Even so, it is still far to humid in the house.Is there a solution to this problem of excessive interior moisture?

The first time we did it, we had the other sides of the shower wall studs all exposed, ran the shower for several hours to watch for any signs of moisture, and saw nothing. 

My impression is that the moisture was visible inside the closet. 

not sure if i mentioned this one... a few months ago the tenant called and said there is some moisture/water dripping from the kitchen cabinet under the sink.i went there and saw the entire inside of the cabinet is wet. it's been dripping for some time. i asked her why she did not call me earlier and she said, "I never use this cabinet". when i looked around, half of the kitchen cabinets were empty.

Did you check her ceiling with a moisture meter?  

I like some of Beth's theme of things that do double duty (like new door mats and bath mats, which are dual purpose gifts that also protect your property from grit and moisture!).

Could be because of something with the downstairs shower and the moisture rising?