I have a Kenmore fridge, manufactured in 01. I realized the drip pan continuously fills itself up.
Water should only drip into the pan when the unit is in a defrost cycle.
he compressor, compressor fan and the evaporator fan will all be off.
If water is continually dripping into it you may have a water leak if you have an ice maker. Also could be the drain tube is partially clogged.

What I'm assuming is that it's the evaporator fan, located next to the drip pan in the back, bottom part of the fridge.
Well actually it is the condenser or compressor fan.
The evaporator fan is the one in the freezer.

Is this the purpose of the fan... to evaporate the drip pan water?
Yes it does help with evaporation, it also cools the compressor and it's coils.
I would fix it ASAP as your unit is probably working too hard trying to get rid of the heat. Also a good idea to clean the coils while you are under there.

Is there a way to know for sure if this is the problem?
One way is remove it and measure it's motor's winding. If open it needs replacing. If it has continuity that does not necessarily mean that it is OK as it's bearings may be shot.

I usually remove it and hook it up to a line cord, plug it in and see if it runs.
Just be careful that you do not get a shock or smacked with the fan blade.
If it runs, then check if there is power at it's connection under the fridge when the compressor is running.
If no power your problem is elsewhere.