Ok,
Well obviously no-one in the past has been recording it so we need indications of indications and hints which are found in sediment cores of the ocean floors. Anything what can be measured is measured, biota remains, macro fossils of foraminifera, alkenone contents of the muds, ratios of elements, and isotopes, isotopes of everything you can think of.
Alleged tell tale signs of changes in the thermohaline current is changes in geographic spread of a thousands species of foraminifera,
Furthermore, a fast flow evens out the temperature gradient with lattitude, as the equatorial heat is more quickly transported to the poles. So rates of changes in sea surface temperatures can be used to measure this speed. Sea surface temperatures are decuded from isotope ratios, element ratios (Mg Ca Sr) and alkenone ratios.
So thermohaline current flows are basically deductions of deductions. It may be clear that this is very tricky, since sometimes other events may cause the same type of remaining evidence.