An interesting problem. From the physics, I would assume that the cable length gives you the fundamental frequency (length = wavelength/2) and cable strumming would show up as harmonic lines. However, you may need to down-sample your data significantly. Taking you figure as an example, I would down-sample to 10 Hz bandwidth, and look for harmonic lines.
Caveat: I never have done it, but this is where I would start.
Edit: Taking Urick (1983) from the shelf one finds on Page 370 a better explanation:
This form of self-noise occurs in a current of water, and is the result of cable vibration induced by the eddies or vortices shed by the cable. This is the "aeolian harp" effect, or the singing of telephone wires in the wind, that has long been known in air acoustics. The frequency of vortex shedding is given by the simple expression f = S * v / d, where S is the dimensionless "Strouhal number," v is the water current speed, and d is the cable diameter in the units of length of v. The Strouhal number happens to be a constant equal to 0.18 over much of the range of current speeds and cable sizes occurring in water. This with a 1-cm-diameter cable in a 1-knot current (51.5 cm/s), the strumming frequency will be 9 Hz. Strumming noise can be readily alleviated by a number of means, including using a faired cable, keeping the natural frequency of the cable vibration well separated from the strumming frequency, isolating the hydrophone from the cable (as by such simple means as suspending it from the cable by rubber bands), and employing a hydrophone having an acceleration cancelling design.
I hope this quotation helphelps for those who have no access to Urick.