Why is this box constantly tripping?
The advice given in the other answers is good: begin by assuming that the safety system is operating correctly. Find the fault, and hire an electrician to do so if you cannot. Only if you have evidence that there is no fault should you assume that the safety system is broken.

However, it is important to understand what this safety system is for, how it works, and what it does and does not protect.

There are a number of commonly-encountered electrical safety devices found in private residences; the three most common are:


ground fault protection. That's probably what this device is.
overcurrent protection. It is possible that this device also provides overcurrent protection, but not all ground fault protectors do.
arc fault protection. It is unlikely that this device is an arc fault protector.


Each solves a different problem, and you cannot rely on a device that solves one problem to solve the other unless the device is specifically designed to do the functions of two different kinds of protection.

The purpose of a ground fault protector is to prevent any current from going from hot to ground through a person, or, in the case of your barn, livestock and so on. Ground fault protection works by monitoring the current on the hot and the current on the neutral. Imagine that electricity is water; a ground fault protection device verifies that the same amount of water is going down the drain as is coming out of the faucet. If the drain is draining less than the faucet is producing, the water has to be going somewhere unexpected. If the ground fault protector detects that the difference between the hot and the neutral currents is more than, say, 30 milliAmps in your case, the hot is turned off by the box tripping.

Note that a ground fault protector provides no protection against electrocution from a hot-to-neutral connection through a person! In that case you are simply an electrical resistance like any other light bulb, and the ground fault protection will not detect the problem. If you somehow manage to touch both the hot and neutral wires of a ground-fault-protected circuit without 30 mA of the current going to ground, the device will not trip.

If your box is tripping frequently because of a ground fault then either (1) something is miswired; you have two hots sharing a neutral for example. (See below.) Or (2) something is draining current from hot to ground somewhere in the system; if you get between the hot leak and the ground, you're going to be the path to ground. Find it and fix it before that happens.

The fact that it happens more when things are wet is a sign that this is a ground fault. Something is draining current from hot to ground, but it is a good enough insulator to not trip the box when it is dry.

A ground fault protector does not protect against fires. There are two ways that electrical fires typically start: either the current is too high for a sustained period, and the wires heat up enough to burn the walls, or the current is too high for a split second, an arc is produced by the electrical outlet / the frayed lamp cord / the loose connection / whatever, and the arc makes sparks that light something on fire.

Ground fault protectors do not protect against either fire-causing failure. It is entirely possible for an electrical system to be way over-current but the current is still equal on both the hot and the neutral. And arcs might be so fast that the ground fault protector does not have time to trip before the current imbalance is restored.

Overcurrent protection by contrast typically monitors the current on the hot only. If the current on the hot exceeds some threshhold, the current is interrupted. Overcurrent protection does not protect against ground faults; if you have a 30 amp breaker and you are draining one amp of current to ground via your body, the 30 amp breaker cares not a bit. The point of the overcurrent protector is to prevent a short circuit from raising the current on the hot to the point where it could start a fire in the walls; it takes way less current than that to kill someone.

Overcurrent protection also does not monitor current on the neutral. The current on the neutral and the current on the hot are supposed to be equal -- recall our water analogy. But imagine you had two faucets pouring into one drain. The current on the two hots might be below 30 amps, but the combined current on the neutral might be higher. Just because the neutral doesn't have voltage on it does not mean it does not have current. It can still get hot enough to burn. Never share a neutral between two hots; that's an "Edison circuit" and they are not safe. (And they will not work with a GFCI, obviously, since the currents on the hot and the neutral are almost never equal if any current is flowing.)

Similarly, arc fault protection does not protect against you being the path to ground for a low-but-deadly current, or sustained high currents; it protects against transient high currents producing arcs that could set nearby paper or carpet or mouse nests or whatever on fire.