How can I bore a hole through a framing member, without causing a lot of damage to the wall covering?
Best way to work through this is with a flexible drill bit.  Most common length is 6'. You can either drill down from your hole or up from your basement. There is a tool to help you guide the bits:


(source: homedepot.com)
   
(source: homedepot.com)  

If 6' is not long enough you can buy bit extensions.  Most of the bits have a small hole at the end of the bit you can use to attach a pull string once the bit is drilled through to its destination; this making pulling a wire back through relatively easy.  They are also usually self-tapping so you just aim it where you want and let the drill do the rest of the work.

Make sure to mark any important infrastructure like power and gas lines - you don't want to hit any of those while drilling.