Deep Space Network sites in California, Madrid and Canberra perform an excellent job, for example receiving from Voyager 2 (16+ billion km) at -150 dBm power. However, would it be more advantageous to operate sites in earth orbit ?
Let's think about what it would take to do what you're suggesting.
You could put it in LEO (because that's much closer and cheaper), but then you have to deal with the Earth occluding it frequently. You could put it at the Earth-Moon L2 point (much more expensive), where the moon blocks signals coming from the Earth, as mentioned later, but then you have have to relay all communication to it because the moon is in the way, adding a lot of complexity to the system, especially since you'll want more than one antenna for the purpose of supporting multiple missions.
Once you've got it there, you also have to deal with a more noisy environment (radio spectrum-wise) than on the ground. From wikipedia:
Each facility is situated in semi-mountainous, bowl-shaped terrain to help shield against radio frequency interference.
Where in space you're getting bombarded by a broad spectrum of energy from the sun, as well as having line of sight to an awful lot of the surface of the Earth, some of which will have transmitters pointed up, particularly your own communication system, because what good is a communication system in space that can't talk to the ground.
Also, when you look into it there are frequencies at which the atmosphere is transparent, so being outside the atmosphere doesn't make a big difference given the right frequencies used: (Source: Australian Space Academy)
Long story short, no: you'd spend a lot of money building and launching huge radio antennas into orbit, and once there (where they're much harder to maintain and particularly vulnerable to collisions with space debris due to their size) the environment isn't actually more conducive to communicating with distant satellites anyway.
I would say that no, an orbital DSN would not be more advantageous. The main problem with receiving transmissions is due to the distance they are being sent from. The DSN antennas have to be very sensitive because the transmission power falls off with the square of the distance. Putting the antennas in space would not make them more sensitive.
If we wanted to place these receivers in space, they would still have to be incredibly large. It would be very difficult to build and launch such large satellites. Once they were in orbit, it would also be very difficult to service or upgrade them.
Secondly, the whole reason we have the DSN is to maintain communication with space probes no matter which direction the Earth turns. If we used satellites to receive these messages, they would then have to be transferred back down to the Earth. So we would just have another link in the system. It's easier just to have them transferred down to Earth directly.