Checking the drain tube from the freezer for a clog and checking the drip pan itself for damage are the two steps for resolving an overflowing drip pan under a freezer. Normally, water in the drip pan evaporates before the pan overflows.
If a drain clogs inside the freezer, water can overflow. Taking all of the contents of the freezer (and refrigerator, if it is a combined unit) and wiping the drain out with a solution of soapy warm water combine to clean out the top of the drain, letting water run through more freely.
When food debris makes it further down into the drain tube, or excess water from the food compartment freezes, the line can clog. If this happens, sending a vinegar and warm water mixture down the tube with a turkey baster should clear out the drain. A pipe cleaner can dislodge debris as well.
While a leaky drain pan is not going to overflow, it also sends water out onto the floor. If water is on the floor beneath the freezer, but the drain tubes are not clogged, drain pan damage is the likely culprit. Removing the grille and pulling out the drain pan are easy steps that reveal damage quickly, and the replacement process simply involves taking the old pan out and putting a new one in.