How is/was continental drift monitored?
Scientific GPS and SLR have been used for some time now, and the measurements are rather accurate. Not only do we measure horizontal movements of tectonic plates, but also uplift as e.g. in the Tibetan and Colorado plateaus. 

Before the GPS was introduced, paleomagnetism were used in some studies to estimate the velocities; The age of oceanic crust are estimated by various methods and the travel distance is estimated from the width of the magnetic polarized bands in the ocean floor. To understand paleogeography and make good models we need to incorporate all branches of geology. E.g. fossil records, paleomagnetism, radiometric age estimation of rocks. 

Some active faults (E.g. San Andreas) can also be directly measured with laser ranging or, in some cases, with a measurement tape to measure deformation of trenches or railway tracks. 

Still, scientists are trying to understand the exact driving forces behind plate tectonics. Modelling of mantel dynamics is an interesting field of research and many questions remains.