If the drain pan has cracks and is leaking, replacement is the best option, but if it is overfilling, checking the lines running into the drain pain is the best repair solution for the problem. Normally, a drain pan empties via evaporation, but drainage problems can cause overflow.
Start with the drains inside the freezer and refrigerator compartments. Take out the food and other contents, and use a solution of soapy, warm water to scrub out the drain. Ice and food particles can cause these drains to back up, and cleaning it can alleviate this problem.
Another possibility is a clogged drain line running form the food compartment down to the drain pan. When this line is clogged, water that drains from the food compartment can turn to ice and block the line. Sending a warm mixture of vinegar and water down the drain tubes with a turkey baster may resolve the blockage and leave the tube clear. Alternatively, use a pipe cleaner to clear the first several inches of the tube.
If the drains and tubes are clear, pull the drain pan out and look for damage. Generally, split or cracked drain pans need replacement rather than repair. If there is no damage, check out the condenser fan near the drain pan. If the fan has stopped, but the pan is full of water, the motor on the fan may need to be replaced.