You're lucky to be starting with an unfinished basement and exposed rafters!
Yes, absolutely use all the space you can, the more volume the better. I recently moved from a 1600 cu ft space to a 4200 cu ft space and the difference is amazing.
I would probably
not put in a raised floor, which can cause new acoustical problems if it isn't ultra-solid, plus subtracts height from your already too-low ceiling. Put in a carpet instead to get your equipment out of physical contact with the floor.
Don't put in a drop ceiling. Just fill the space between the floor joists with insulation and cover with conventional sheetrock. Be sure to leave removable panels anywhere you have electrical junction boxes or drain cleanouts.
A big + to previous comments regarding grounding. Install a single dedicated circuit to power your gear. Wire it all the way back to the service entrance - don't tap off an existing upstairs circuit. You can put your lights on a separate circuit, but you want all your audio gear plugged into the same circuit and sharing a common ground.
Note that the National Electric Code is concerned entirely about safety. When it comes to grounding, all they care about is that the breaker activate fast enough if there's a short. I have gone round and round with more than one electrician thumping the NEC book like a fire-and-brimstone preacher and his bible. Truth is, the NEC just doesn't address grounding issues as they pertain to computer and audio equipment functioning properly.
The easiest plan is to run a single circuit down to the basement, wired directly back to the service panel, the ground and neutral wires touching nowhere except at the neutral-ground bond in the service panel itself. Then plug all your equipment into that outlet. It's OK to add a power distribution box (fancy extension cord) from the outlet to your rack, just makes sure to plug everything into that box.
Your biggest issue in the basement will be bass trapping. I assume the walls are concrete as well as the floor. Poured concrete doesn't absorb much sound! Bass frequencies will continue to bounce around in there for a looong time. Fortunately, you have a lot of space. Get some rolls of regular fiberglass insulation and stack them in the corners. You don't even need to take them out of their plastic bags, just stack 'em up. You can cover them for appearance's sake if you like.
Then all you need are some absorbers along the walls and you should be able to create a very nice recording space.
One more thing: before you do
anything, spend 25 bucks on Rod Gervais' book
"Home Recording Studio: Build it Like the Pros". Rod isn't just an expert on acoustics, he's also a builder with lots of practical experience in construction.