The basement is different than the rest of the house because the floor is colder, attracting more moisture, and the cold air falls and is trapped on the floor as well. So the first step is to plan for the exhaust fan to draw the air off the floor of that basement bathroom, not off the ceiling as in the rest of the house.
Then you need to find a way to cut through to the outdoors. Most contractors will cut right through the concrete simply because it does not mess up any siding on the outside. Most DIYers will cut through the header section above the basement wall, because it is wood -- even if they then have to cut through the siding. Both routes work well as long as the outside hood is sealed against water penetration. If you have a window that will not be used you could replace one section of glass with plywood and go through the window -- an easy route, but not the most elegant.
Then if you put a dis-humidistat on the fan control so that it will run whenever it is too humid in the basement, you will get good basement moisture control. At least two companies have now made special units that do all of the above -- the one in the graphic is from the company Humidex. These can usually be found in renovation centres.