At idle with no accessories on your voltage should be well in the 14V area with out needing the wire upgrade. Also what is the batt voltage with the car off? You should see a 1.5V-2V difference with the car on/off. Now a couple of things:
1- your factory wires may be turning green, make sure your terminals are free of corrosion and the ends of the cables are good to. Best way to check this is to get the voltage at the ALT. (behind front pass tire) which leads to step two....
2- If the voltage at your ALT is different then at the battery you have a cable problem and you should replace the grounds and positive cable from Batt to ALT. If they are the same...
3- start with a battery charger, use the lowest 12V setting (like 1A or .5A) and charge the battery over night. It may be wise to disconnect the battery ground in case you have a other issues in the car (key off battery drain etc) Check the voltage in the morning, both with charger on and once after disconnected from charger, be sure to give the battery 10 minutes or so to stabilize at one constant voltage.
At this point adding a cap will make no difference, you have a problem, adding a cap is not going to solve. Once you find the problem (battery, ALT or cables) you can then fix it properly. You will not need a larger alternator, my guess is that if it does turn out to be that, it is probably just a bad regulator or worn brushes. Both go bad over time.