Hello, Best bet is to apply some good quality Tub/Shower caulk against the bottom edge. I know it doesn't look good and it will get dirty looking but the problem is that the water "wicks up" behind the walls and down. Thats unless it is leaking through the grouted areas. This needs to be sealed as well. Try and see if that works.
Also, if you think the tub is moving, can you get behind, under or around it to sure it up some. Maybe apply some spray foam? The tub itself should also be setting on a ledge to stop from bouncing up and down.(2x4 or 1x4) This is suppose to be installed as tub is being set so the tub sets on a firm ledge, then it gets nailed to the wall.