Factual error: [Special Edition] The Statue of Liberty is incorrectly shown to be facing the Narrows bridge.
Factual error: When Bud is dropped onto the spacecraft floor by the alien who rescued him, air trapped in his suit comes off in large bubbles. At 18,000 feet bubbles would be compressed so far that they would be BB size. Also, with all the movement Bud experienced on the way down, no air could possibly have remained trapped in the folds of his suit.
Factual error: During the flooding of the drill rig, Bud's wedding ring saves the day by preventing the barrier door from closing. Even titanium couldn't stop a moving hydraulic barrier, in ring form. It's very hard in terms of resisting scratches, but that doesn't mean it wouldn't deform or crack.
Factual error: When disarming the nuke Bud is trying to figure out which one of the two wires to cut. With only two wires running between two separated modules (in the film, one part is unscrewed from the other) any current flowing through one wire must be returned through the other in order to conserved electric charge. Cutting either wire, in this case, will result in breaking the circuit. There is no way for either side of the circuit to detect which wire(s) were cut.
Answer: At the depth you're asking about, the water pressure is not enough to crush a diver's unprotected body (which is mostly solid/liquid and thus relatively hard to compress). The only thing they have to worry about is their lungs (gases, and therefore things containing gases, are much easier to compress). This worry is addressed by the pressure and composition of the air that they breathe on the rig, in their submersibles, and in their air tanks - this air has different proportions of gases (especially oxygen) compared to surface air, and is also at a significantly higher pressure - high enough to counter the pressure of the water that deep. (Side note: another way you can tell the atmosphere they are breathing in the rig is set at the same pressure as the water at that depth is that there is open water in the moon pool - and it's not rushing in to fill the entire rig, which it would if the air pressure was less than the water pressure).
Aerinah