[Flightdeck]

When you say they can maneuver the elevator, you're implying that the anchor isn't firmly attached to the earth's crust. The force pulling up on the anchor would have to be massive enough to lift any "maneuverable" ship out of the water, although I don't know about the oil platforms

[Brett66]

The force pulling up on it will be roughly balanced with the weight of the cable, with allowances for payload on the cable. It won't be near enough to lift a ship of several thousand tons.

[RightWhale]

The force pulling up on the anchor

That is correct. The entire structure would be in tension, so the heavier the structure and elevators [assuming multiple elevators so they can lift enough cargo to make it worthwhile] the greater the tug on the anchor. You'd eventually want something the weight of the Cheops pyramid at the ground end.

[Atlas Sneezed]

The force pulling up on the anchor would have to be massive enough to lift any "maneuverable" ship out of the water, although I don't know about the oil platforms.

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No, the force would only need to be enough to support the weight of the maximum lifted payload at any one time (the payload weight approaches weightlessness as approaches geosynch.)

With a maximum load being lifted, the tension would need only to be nominal. Simply put, the "ship" would need to weigh as much as the max payload. That's not hard to do with floating concrete ballast/barges.