Assuming the beams are white-light beams, the answer is "no", the two can't be distinguished from each other. In the original form of this answer, I wrote:
If they are coherent monochromatic beams, then they can be
distinguished from each other by analyzing their polarizations. Beam
A would be vertically polarized, while Beam B would be polarized
circularly, elliptically, or linearly at 45 degrees.
which is not really responsive to the question. More accurately:
A coherent monochromatic beam has an unchanging polarization state: circular, elliptical, or linear. It can't really be said to be unpolarized, because by using an appropriate combination of wave plates it can be transformed to any desired polarization state. So, Beam B and Beam A are effectively the same: feed Beam B through a suitable combination of wave plates and it will be identical to Beam A. The details of how the vertical and horizontal (linear) components of B are combined, and the polarization state of A, will determine what the combination of wave plates needs to be. In other words, there is not a way to distinguish the two beams.
A beam that does not have a definite polarization state must have a range of frequency components, with different frequency components having different polarizations. Then, the net polarization in the beam at any instant is effectively random -- but then the beam is not monochromatic or coherent.
The same question applies for a single photon : Is it possible to distinguish a unpolarized single photon from a H or V polarized photon ?
This is a bit different, because a single photon can only be measured once, whereas a beam can be measured many times. For a single photon the answer is "no".