They basically used the same method in use since ancient times - lower a weight on a the end of a long string and pay it out until it appears to stop sinking. To do this most accurately they used a thin wire (rather than rope or string) so as to reduce the effects from currents pushing the wire sideways and resulting in an artificially deep measurement. This was the standard method for measuring ocean depth until acoustic (echo sounding) methods were developed in the 20th century. It's one of the reasons deep ocean measurements prior to 1900 hundred are so rare - a single depth measurement could require an entire day of labor.

more at: http://library.ucsd.edu/dc/object/bb8021588p

edit: fact checking my own statement - according to The Oceans by Sverdrup, Johnson and Fleming, the sounding equipment used on Challenger was a hemp rope - use of wire was introduced the following decade.