While @DavidHammen's answer is insightful and informative, I'll add to the mix by addressing specifically:
What instruments and/or equipment will be used to detect frame dragging?
From this JPL Juno press kit's Science overview:
To measure these tiny shifts, Juno’s telecommunication system is equipped with a radio transponder that operates in the X band, which are radio signals with a wavelength of three centimeters. The transponder detects signals sent from NASA’s Deep Space Network on Earth and immediately sends a signal in return. The small changes in the signal’s frequency tell us how much Juno has shifted due to variations in Jupiter’s gravity. For added accuracy, the telecommunication system also has a Ka- band translator system, which does a similar job, but at radio wavelengths of one centimeter. One of the antennas of NASA's Deep Space Network located in Goldstone, California, has been fitted to send and receive signals at both radio bands. An instrument called the Advanced Water Vapor Radiometer helps to isolate the signal from interference caused by Earth’s atmosphere.
JPL provided the Juno telecom system. The Italian Space Agency contributed the Ka-band translator system.
So it's two transmitters and two receivers plus a water vapor radiometer on the ground as well as two separate transponders on Juno using different bands that work together to try to overcome the challenges of measuring the tiny effects outlined in @DavidHammen's answer.
The idea is that signals in the two transponder bands will be delayed by different amounts by water vapor in the Earth's atmosphere, and by measuring the difference between the delays one can approximately subtract the effect of the water on the round-trip light time. Some GPS/GNSS systems apply two-frequency measurements to reduce GPS error due to water vapor, though there are other techniques they can use as well.
For more on this see:
Original answer (and still helpful)
I'll just address the "How" part (leaving "How well") to @DavidHammen's answer.
There is no "frame dragging sensor" aboard Juno. As an orbiter of Jupiter, it's trajectory is the "sensor" and the trajectory is determined by various radiometric techniques based on delay-Doppler measurements from the Deep Space Network on Earth.
These can include both two-way (send a coded, stable frequency signal, receive the phase-coherent rebroadcast from the spacecraft, apply correlation to precisely determine the delay and the Doppler shift) and more complex methods. For more on that see @MarkAdler's answer to
For more on precision tracking in general, see @TomSpilker's answer to