Where is the smell? Is it in the unfinished room or the entire basement? Chances are the odor is in something porous, like the carpet/pad. You may have to remove it. That's what I'd do. I know that my dog likes to sleep up against things. Cleaning just the floor might not be enough. It could be in walls of the unfinished portion.

Lastly, didn't you smell it when you went through the house before your purchase? Did your inspector notice it?

Are you sure this is all dog smell? That musty dog-like smell might be caused by humidity/mildew or even a dried-out floor drain. I had a minor sewer gas problem that didn't smell like sewage. Just really musty.
In any case, you may need to get rid of the carpet entirely. Some smells just don't come out.
You could also try sprays formulated to get rid of pet odours, and even spray some unlikely surfaces, such as exposed wooden ceiling beams, that might have soaked up the funky smell.

Get some pet odor eliminator.
There should be a floor drain somewhere. MAke sure the trap is full of water. It may even be under the carpet, but is usually near the washing maching connection.
Concrete can absorb a lot of smells from pets. It may require treatment with an odor eliminator, followed by sealing with dri-lock or another concrete sealer.
The carpet may be ruined. If the smell is strong tear it out.