Like Swirl-girl asked, if you can post a picture taken from ground level that would help a great deal.
Right now, it looks like you've got a hot tub where the smaller green square is, which would be why you can't build a basement there. Also, you do have a basement in the larger green square - it appears to be a room on basement level 2. Are you trying to use the basement tool at this level (that is, the same view you took this picture)? The basement tool creates rooms under the active level, not on it. You'd have to be one level up, at the grass/ground level, and then you should be able to draw your basement rooms using the tool.
Personally I've never seen anything other than ungridded black space around a basement room in my games, so it looks perfectly normal to me.