How can I safely pilot the Cyclops?

Cyclops with engines off and lights (internal and external) off is completely invisible to all Leviathan class enemies. A Reaper may bump into you and push you around, but you won't take any damage. The only ways how you can take damage with engines off is either the attack began before you switched the engines off (the enemy turned red on the sonar display) or you got accidentally hit by Sea Dragon's fire as it was attacking a lava lizard or such. Even then the damage received that way is minor and easily repaired. Note - not just "engines spinned down" but completely off - click the propeller icon.

Even with engines on, set on slow ahead, and "rig for silent" active, if you remain immobile with the leviathan anywhere but in nearly immediate vicinity (inner circle on the sonar) you won't get attacked. And if the leviathan is anywhere farther - outskirts of the sonar or outside of it - you can move slowly; if the leviathan is still within sonar range, don't use full speed of "slow ahead", just sneak slowly where you want to go.

Note: "Rig for silent" continuously drains power - and with engine off, it's completely pointless, so as soon as you switch engine off, turn the silent mode off; before powering it back up switch silent back on, until you're out of the encounter.

The general tactic is: move at slow ahead, unless you know for a fact there are no leviathan class predators. If you spot one (or hear one nearby), activate "rig for silent". Observe the radar; if the leviathan is approaching, switch engines off, silent off, and wait. When the leviathan moves away, silent on, engines on, and start sneaking. If it's lurking at intermediate distance, stop with engines on, move if it moves away, switch engines off if it's getting really close.

Adhering to this tactic with no errors guarantees 100% safety in all encounters - you can sneak right through the middle of every leviathan's patrol zone reliably. If you make an error though - e.g. fail to deactivate engines, drive into leviathan's view at full ahead, or leave floodlights on, you may get attacked. In this case activating the shield right before the impact is a good solution. Dropping a decoy and moving away (slowly, silently!) while it distracts the leviathan works too. And of course you can switch the engines off, lights off, take the hit, and - the leviathan will quickly lose interest in the unpowered Cyclops. Sneak away, repair using the repair tool.

Let's add - the thermal generator upgrade is invaluable in the deepest parts of the game. Safe evasion of leviathans is best performed with patience, often a lot of waiting, slow movement. Meanwhile, lava larvas attach to your Cyclops and drain its power. With thermal upgrade you'll be recovering power faster than you lose it, even with "rig for silent" active non-stop, and you can play the waiting game against leviathans correctly. If you do have leeches attached to the hull though, and no way to recuperate the energy before the encounter ends, you're in a bind. Retreat, then kill the larva with a knife (just getting them off with a knife, or blasting them away through flicking the shield on, without killing them works for an awfully short time; shooting them with repulsion cannon works a little better, and can be done from inside of Cyclops in some cases - watch the damage display for larva locations) - or take risks.

The different speeds you can select affects the amount of noise you make. The faster your speed, the louder you are, which means the further creatures can hear you. You can engage silent running mode at any speed setting to further reduce noise (by 50% according to the Wiki), but it consumes power at a faster rate. Be sure to use your cameras to be aware of your surroundings as well.

If a creature is coming at you, you can launch a creature decoy to divert them away from you and allowing you to escape. You may also want to use a shield generator to temporarily make the Cyclops invulnerable. The shield generator uses a significant amount power, so be sure to use it only when needed. Lastly, the sonar upgrade can aid you in pointing out creatures you can't see.

Personally, I like to navigate the very deep parts of the game with the PRAWN suit instead. I would take my Cyclops down deep, but would park it in a place that is safe from any large creatures, and continue to explore with the PRAWN. The PRAWN is actually pretty fast once you have the grappling hook upgrade to swing yourself around with, as well as the jet pack upgrade.

Timmy Jim is completely right. I'd like to add my strategy.

My approach to bringing the cyclops into the deep areas has been to use the shield generator, but only toggle it on when an enemy is just about to impact the sub. As long as the shield is powered on during the moment of impact, no damage will be sustained. You'll have to keep a close eye on the radar for incoming threats, but this strategy minimizes power use and damage taken.