To deal with this type of problem, you must be careful to define exactly what system you are dealing with, and then not change that system part way through the problem. This definition allows you to be very clear about whether the "system" has any external forces acting, and thus whether the momentum of the system is constant or not.
In this case, you seem to be defining the wagon itself as the system, but then talk about the wagon as gaining weight, implying that the definition of what constitutes the wagon system is changing.
Let's try this: the system is the wagon itself, without any stray mass that may be added. The wagon has a certain amount of momentum, and since there is no outside force of friction, that momentum is constant.
But then the rain starts to fall. None of this rain is included in the system, even though it gets trapped inside the wagon. But in being trapped, the vertically falling rain also exerts an horizontal force on the system: either impacting the back of the wagon in the air, or hitting the bottom, and flowing towards the back of the wagon. All this means that there is an external force exerted by the rain on the system, and momentum of the system is not conserved.
We can start over: the system now is defined as including the wagon and all the vertically falling water. Since the rain initially has no horizontal velocity, the total momentum of this new system is just that of the wagon.
Now the rain starts hitting the wagon. But under our new definition, all of the rain impacting is an internal force, and cannot change the total momentum. This new system is isolated and momentum is conserved. So now, since more and more of the system is travelling with the wagon, the wagon must slow down. Internally, momentum is being transferred from the wagon part to the rain part of the overall system.
Best Answer
I think we are supposed to assume that the buoyancy force of the balloon is equal to the weight force of the balloon, ladder, and climber. If this is the case, the system is in equilibrium with its environment, with no net forces to the environment.
You could look at it as a center of mass problem. We assume the ladder has negligible mass. When the climber reaches the balloon, they will be together at the center of mass of the system. The climber and balloon start apart by the length of ladder $l$, while the center of mass is $\frac{l\times M}{m + M}$ above the climber, and $\frac{l\times m}{m + M}$ below the balloon. The ratio of the balloon distance to travel compared to the person distance to travel is $\frac{m}{M}$. Therefore the velocity of the balloon at any point in time is $-v_{person} \times \frac{m_{person}}{M_{balloon}}$.
Center of mass problems can be reformulated as conservation of momentum problems. At $t_0$, the balloon and person are both at rest, with the person hanging off the bottom of the ladder - net momentum 0. At $t_1$, the climber has momentum $m_{person}\times v_{person}$, so the momentum of the balloon must balance this: $M_{balloon} \times \frac{-m_{person}\times v_{person}}{M_{balloon}}$. Same result.