Variable mass system confusion

not sure to have understood the question.

Anyway if you are concerned about the conservation of momentum, the sistem Earth-Box gains the one lost by the droplet.

In the vertical direction, your are right: droplets are "balanced" by the normal reaction of the box which stops them without doing any work, so that no noticeable change in momentum of the box is observable in the vertical direction (this is of course a small assumption: if the droplets were very heavy and fast or if the box is very light, then there would be a change in momentum: but for the moment we treat the problem as if the box has infinite mass wrt the individual droplets so that the collision just stops the droplets: all of their velocity just disappears in the collision with the box).

However, once the water is in it has to move together with the box in the horizontal direction. What happens is that (through friction, which pushes the droplets together with the box) the box is lending some of its momentum to the droplets and by doing that is slows down. On the other hand, the droplets gain momentum and start moving with the box. Because the only force acting in the horizontal direction is the one between the box and the water, which is an internal force, conservation of momentum applies and all the momentum the box looses must go into motion of the water.

You can then write that the total initial momentum $m_bv_0$ must equal the final momentum $m_bv_f+Dmv_d$ where $v_0$, $v_f$ and $v_d$ are the speeds of the box at the beginning and end, and the speed of the droplets at the end (which at the beginning was 0, because they were falling down vertically). By a further (reasonable) assumption that the box and the water now move at the same speed, your problem can be solved!