I don't trust Photobucket so I didn't look at the pictures. But to me, it would depend on a few things.
1. Height in the basement. For both recording and mixing. Very low ceilings in typical basements usually require extensive absorption which usually translates into lining the floor above joist cavitys with absorption, which also means, to increase the transmission loss performance of the membrane, also must be lined with a couple layers of drywall. Also, basements are natorious for moisture problems, AND are the defacto place for HVAC units and ducts, which must be delt with for TL as well. Depending on your isolation needs, these can be a real pain in the but to properly address.
vs a blank canvass in a seperate building.
2. If this outbuilding is a typical shed type, depending on the roof construction and your TL requirements, I would use the space at the apex of the roof for a massive superchunk of 3lb density rigid fiberglass or mineral wool
for a bass trap as well as all 4 wall to ceiling/roof corners too. However, this all has to do with your isolation needs vs the environmental noise floor and distance to adjacent neighbors, AND the roof construction(rafter/ties or trusses). There are many ways to increase the transmission loss potential of a blank outbuilding. Unfortunately, the one factor that most older small outbuildings share is a wood framed floor instead of a slab. This one factor alone could make the difference as a wood framed floor is a membrane, which will transmit LF/impact sounds(drums) like...well, a drum head.
And increasing the mass may or may not be possible depending on ...1. The depth of space between the ground and the bottom edge of the joists and 2. the load capacity of the framing itself. However, if the floor is a slab...thats a PLUS.
3. Size. What are your needs? What are the sizes of the 2 options.
4. HVAC. Regardless if there is an existing HVAC in the basement, and depending on your budget/isolation requirements, tieing into existing ductwork creates a flanking path for sound to the upper floor, unless correct plenum/baffles are installed..which may or may not be possible. Not only that, but STRUCTURAL transmission paths must be addressed as well...again..depending on your isolation needs and budget. Bottom line is this. Anytime you contain sound within an envelope, you now have to absorb it...especially low frequencys. AND, you either need lots of mass or a decoupling strategy to keep LF from transmitting throughout the house.
vs...IF, the distance to your neighbors is far enough and the environmental noise is low enough vs the SPL created within the studio, the outbuilding may not require extensive mass for isolation..which translates into LF sound transmitting to the outside world..which also translates into less cost and less LF treatment within. In other words, what sound escapes does not have to be absorbed. Another plus.
On the minus side, the outbuilding may require an external HVAC and ventilation paths. If you insulate this building correctly, heat buildup will occur rapidly. People and equipment generate heat. Not only that, but even musicians like to breath.
In order to make a space relatively SOUNDPROOF, you also must make it AIRPROOF, which creates a conundrum. After spending mucho time and money to build an isolated building envelope, now you must penetrate this envelope for ventilation, not to mention HVAC...not an easy task to keep the iso shell TL performance target, and satisfy breathing/comfort levels at the same time.
Anyway, thought I would share some of my decision making thoughts with you. In the end, only you know all the potentials of your project. Like budget, skills, existing conditions etc etc. Personally, I'd sit down and make a list of each and every element of your project, and juxtaposition them on each option, and then compare all the possible ramifications of each and their solutions. Then you have a comprehensive decision hiarchy to work with.
fitZ
Like I said, thats a definite plus.
Problem is though, if you just rush it, and don't plan things out, you run the risk of completely negating all your work and money. "Soundproofing"