I'm not familiar with land-based methods, but for global measurements, one method is to use satellite altimetry (I'm more familiar with the geodesy side, but many of the same satellites are used). I think many of the current methods interpolate global or regional currents from a sparse network of buoys. As more radar satellites are launched, however, satellite measurements of sea surface currents will become more common.
Global-scale satellite-based radar altimetry measures the average elevation of the ocean's surface over a few kilometers (the exact amount depends on the wavelength of the radar band being used). Over this distance, waves cancel out, so the measurement is accurate to within a few centimeters. (This is aided by using the motion of the satellite to enhance the image, referred to as Synthetic Aperture Radar (SAR).)
Surface currents in the ocean are directly related to the slope of the ocean's surface. (Again, at the scales we're working with, wind-formed waves cancel out.)
Once you have an accurate snapshot of the elevation of the ocean, you can calculate the direction and magnitude of the surface currents. (Note that this is only the surface currents! Deep ocean currents are different matter altogether.)
This is also how we coarsely map the ocean floor from space. (For example, Sandwell & Smith's extensive work: http://topex.ucsd.edu/WWW_html/mar_topo.html)
The average elevation of the ocean's surface with respect to center of the Earth follows the geoid (by definition).
Seamounts on the ocean floor cause a preturbation in the geoid above them. Water effectively "bunches up" around seamounts (or rather, the surface of gravitational equipotential "bunches up" above the seamount).
By repeatedly measuring the oceans surface (over several years) we can remove the effect of surface currents and get an accurate picture of what the geoid looks like over the ocean. You can then use this to predict water depths over the oceans on a spatial scale of ~1 kilometer.
In fact, to calculate surface currents, you need this information to begin with. The surface currents are calculated as deviations from this measurement of the geoid.