In The Long Utopia (Pratchett & Baxter) there's a whole arc about an alien race speeding up the rotation of Earth (well, a version of Earth). The method used is a Planetary Spin Motor, as described by Freeman Dyson. It's a bit involved but ultimately it's transferring kinetic energy from a bunch of conductive masses thrown into the planet's magnetic field and applying it to the planet as rotation. Changing the direction of transfer - by altering the trajectory of the masses. This would extract rotational energy as velocity in your masses, which can then be converted to electrical energy by passing them through a series of coils that bleed away the kinetic energy.
Something similar could be achieved by tidal coupling rather than electromagnetic couple. The Moon is tidally coupled to the Earth and as a result the Earth's rotational energy is transferred to the Moon. The result is the Earth's rotation is slowing down and the Moon's momentum is being increased, raising its' orbit slowly. Very, very slowly.
Back to magnetic coupling, David Brin published a story called Tank Farm Dynamo which involved an orbiting double platform joined by tethers. As the station orbits the Earth it passes through the magnetic field, which induces current flow in the tethers between the two platforms. Here's an except from the story:
For an instant I saw the Earth not as a broad vague mass overhead, but as a spinning globe of rock, rushing air, and water, of molten core and invisible fields, reaching out to grapple with the tides that filled space. It was eerie. I could almost feel the Tank Farm, like a double-ended kite, coursing through those invisible fields, its tethers cutting the lines of force -- like the slowly turning bushings of a dynamo.
That was what young Emily Testa had compared it to. A dynamo. We could draw power from our motion if we ever had to -- buying electricity and paying for it in orbital momentum. It was a solution in search of a problem, for we already had all the power we needed.
As the POV character points out though, they had plenty of power. Turns out that what they really needed was to find a way to increase their orbit, not decrease it. You can always hook up solar sails or simply use thrusters to increase your momentum, then use the magnetic coupling to the Earth to turn that momentum back to electrical power. Each watt you draw comes directly from the rotational energy of the planet.