We need to ask ourselves... how is a 12v battery going to prevent a voltage drop on a 14 volt electrical system?
With any real current draw from your stereo, voltage drops are going to happen. The goal is to reduce their severity or at least make the drop more of a smooth slope. As stated already, upgrading the Big 3 wires under you hood will help to improve current flow across your entire electrical system. However, simply switching to a deep cycle battery isn't going to provide you with any more power than your current battery, nor will it somehow keep the voltage up at 14v. The only thing a deep cycle is really good for is draining/recharging multiple times. It operates at 12v just like a standard lead-acid battery; therefore, it alone will not prevent your voltage from dropping or headlights from dimming.
Obviously, the heart of your electrical system is the alternator. It provides the 14v and the current necessary to power your system. It may not be the most cost effective upgrade, but if the goal is to actually prevent voltage drops, then upgrading to a high-output alt should be your next consideration. You can spend $150 on a deep cycle battery and still get the voltage drops, or you can spend $200-300 on a new alternator and rest assured that you have helped to solve the problem, rather than merely putting a VERY expensive band-aid over it. And when it comes to installing an additional battery, you're just placing another load on your stock alternator which already is having trouble keeping up.