Alright so I'm big on listening to music in my car. However there's one thing, I can't listen to music (with bass) without the grips on the door rattling. It's only on the driver's door and the passenger door, if I reach over and grab them the rattling stops. Is there any way I can "fasten" them, or somehow rig them to stay pushed down in place?
Rattling has to do with loose parts. The bass frequencies are causing the parts to vibrate against each other.
You will need to isolate the rattle, and tighten the part making the noise. Many times it just needs a little adjustment.
If the part is unable to be tightened any farther, you may need to add some type of buffer material, like thin foam padding. It may be advantageous to take the door panels off and inspect for any other loose parts. I've had to use this method on door handles, luggage racks, side panels, etc.
In the case where the vibration is extreme, some sound deadening material, like Dynamat, may be helpful on the inside of the door panel itself to help reduce vibration in the sheet metal.
Usually, these armrests have fixings that go in from underneath - so the screws may be covered by plastic discs, all you may need to do is tighten them.
In my experience, the parts in vehicles that rattle are usually the ones that aren't screwed in.
Your solution options depend on how much money/effort you're willing to spend. I've taken old garbage receipts/papers, folded them, and shoved them into the cracks between rattling pieces in hidden nooks as cheap fixes before. Alternatively, you can buy thin slices/ribbons of foam and/or rubber to create a buffer between the rattling pieces, or put tape or felt between them.
Often times you don't actually have to do a good job of isolating/insulating the vibrating pieces, you just have to cram something in there to get them to stop moving.