This is a spot where cracks often develop. After cleaning with a a mildewcide, check carefully for a crack. If the grout is cracked, open the crack up a little wider with a grout saw (inexpensive at a hardware store) or steel blade and fill it with a color matched caulk or sanded sealer from the tile store. If the crack has moisture, let it dry well before caulking.

Almost any mildewicide will do the trick.  Grout sealants all seem to work but I would go specifically to a tile shop to see what they recommend.  I think most sealers will work but application and prep are key.