I'll write it out to be clear:
You have two "power wires" connected to your amp. You have a main power wire, (the large awg cable), and a second power "switch wire", the remote turn on / aux turn on / amp bi-wire, that is usually a blue wire with a white stripe. That remote wire is just for the smallest of switches to turn on your amp, (think of it like in a house, it would be the power button on your remote and the main (awg) wire is the plug int to your outlet).
That blue w/white stripe wire is there to turn your amp on, only when your head unit is on, (because what is the point of having your amp on if your head unit is off).
IF that is actually what you want, then you would wire that to a 12v source, and you could add your own switch (toggle, or push button) in line. You run the risk of leaving it on, and draining your battery after your vehicle is off, because you would have a draw. If you are going to do this, you should at least get a switch that has a light on it, so you have a way to be notified.
WHAT I THINK YOUR REAL PROBLEM IS: Why is your amp connected to your alternator? Your alternator recharges your battery, and nothing else. You might be drawing more power than your alt can handle just by the amp, depending on your setup. Your speakers draw through the amp, the power that they need. If you are wired only directly to the alt, you might be drawing more than your alt can handle, and not only could be causing it to get sluggish, or create a larger draw, (that's really just a guess on what it would do, that's not the way you should be doing it), but you could be causing short term, and long term premature failure.
Your amp should be connected to the battery. It will draw what it needs, and your alt will recharge the battery and keep up as it can. If your amp is not wired directly to the battery, that is where you need to start. Your priority should be your vehicle running right, and then your amp.
If after you do that, you should do the big 3 if you don't have that.
Last, instead of wiring your amp that way, with an on or off switch, a better option would be to have a remote knob/bass knob/potentiometer. Even if the amp doesn't have a factory connection for one, you can get a level knob that will plug inline in your RCAs, something like this:
This way you can still have a little bass, just turn it down, so you aren't drawing max power and your battery can build back up, or you can also crank this all the way down, and it will turn the amp signal off so there is no draw. This would give you a cheap, easy, removable (if you wanted to) fix, but you will probably want this.
I still don't get why you would want your amp on, when your vehicle is off. If your head unit is off, then your amp won't play. Your amp SHOULD be connected correctly, and then you would have to wire your head unit's power wire differently. Just like your amp, it has a 12v power wire that feeds it, and an accessory wire that switches it on, when your key is on or accessory. If you really want to do that with your vehicle off, you would wire the switch/accessory wire also to power. You then run the risk of leaving it on and draining your battery if you don't physically turn it off, EVERY TIME, as opposed to what you do now, and just take the keys out.
