Submarines are not really used to measure ocean depth. There are very few that can go deeper than a few hundred or few thousand feet. Ocean mapping and depth sounding(sound as in body of water, not what you hear) is mostly done with sonar.
You send a sound signal down and see how long it takes for it to bounce back. Do a little math to account for the speed of sound through the water and you can get a pretty accurate reading. You can even use it to map the seafloor to pretty high resolution.
Prior to the development of that kind of technology in the 20th century, depth was determined by what was was called a lead line or sounding line. It is exactly what it sounds like. A long line with a heavy weight of lead, or something, attached to the end that causes the line to sink. This line would be regularly marked. As the line sunk you could count the number of marks and get a decent estimate of depth. A fathom, the unit historically used to measure depth, is based on the approximate length of a person's outstretched arms. Which makes sense if you imagine someone laying out rope and counting lengths to determine depth.
The technology is at least as old as ancient Greece and Rome and there are references in ancient works like the Bible and others. The Greeks and Romans had a good idea of the depth of the surrounding seas, including the mediterranean, especially in coastal areas. By taking repeated measurements they could even put together a rough map of the sea floor. This could be used for navigation, to aid in fishing, or just for curiosity. A "tallow cup" was sometimes used to bring back up samples of the seafloor for study.
Sounding lines are most accurate in shallow water, where you can be more sure the line goes straight down. The deeper you go, the more likely that the rope is not straight which will affect your measurements. However, the deepest part of the ocean, the Mariana Trench, was first discovered in 1877 using this method. By the end of the 19th century you even had mechanized sounding lines that automated the process.
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