Basement walls above grade are wet outdoors
There could be two issues: 1) moisture dripping off the building paper under the siding, or 2) moisture entering the masonry cells. 

1) We install building paper (moisture barrier) under exterior siding because we know the siding will leak. Well, that moisture needs to run out at the bottom of the wall. If the siding is leaking really badly above this area for some reason, all that moisture could run out long after a storm and “wet” the wall. 

2) If you look closely at the interior block wall, you’ll notice some discoloring (darker areas) aligned vertically. This indicates moisture is getting into the cells and filling with water. (Masonry blocks are hollow and often not filled with grout when the wall is constructed.)  This allows moisture in the ground to seep into the concrete block cells. (Yes, masonry block is porous.)

Also, if the window in the first picture is the same window in the interior picture, then these moisture spots align perfectly. This could mean the cell is “full” of water.  

Heat will “draw” moisture out of a wall. (That’s why paint on south facing walls blister more than other walls.) So, where the cells have moisture, the sun could draw the moisture out and cause the discoloration. 

I’d:  1) check to see if moisture is running out from behind the siding by rolling a few paper towels up and stuffing them up under the bottom of your siding at the wet area AND at the dry areas. Then compare after a storm, 2) Dig down a few feet at the wet spot and see if the moisture barrier on the outside of the basement wall is damaged. I’d also make sure the material (soil) is porous so the moisture can “flow” away from your wall. If it’s not, I’d replace with drain rock. (Round river rock without fines...no rocks smaller than 1/2” dia.) Worse case: you’ll need to install a French drain to carry the moisture away from the wall. 

Regardless of where the moisture is coming from, I would NOT finish the interior until you find and stop the moisture because it will cause mold.

...but I don’t understand what is happening at the top of your wall inside. Is that insulation?