Continuously? No, for several reasons.
Most significantly, it just isn't possible. Both Voyagers are now so far away that only the 70-meter antennas of the Deep Space Network can communicate with them. Each DSN site only has one of these antennas, and the two Voyagers are both on the same side of the Earth, so there are times when only one antenna is available for the two of them. That antenna can only point in one direction, so except in the rare case of multiple spacecraft close together, it can only talk to one at a time*.
There also isn't any need. The round-trip signal time for Voyager 1 is around 44 hours; for Voyager 2, it's 36 hours. Both spacecraft are pretty much in the middle of nowhere, doing nothing but recording data and periodically forwarding it to Earth. There's no expectation of anything surprising happening, so losing a few hours of data if something goes wrong is no big deal.
Both spacecraft are routinely monitored, but this takes the form of periodic, scheduled communication sessions rather than continuous monitoring.
*Right now, antenna 25 at Goldstone is talking to Mars Odyssey, MAVEN, and the Trace Gas Orbiter, all in orbit around Mars.