Opening the Cabinet
To repair the basket of your washer, you need to first open up the cabinet of the machine. Slide a putty knife under each side of the cabinet top to release the clips holding it down. GE washers will lift up on their hinges, allowing you to access the tub on the inside of the machine.
Removing the Agitator
A broken agitator or faulty agitator seals may cause the basket assembly to come loose. GE washers have one of two agitator types. One agitator lifts up out of the tub without any loosening. Grip the agitator tight and pull up hard to remove it. The second type requires that you remove a bolt. Pull up on the agitator cap to expose the bolt underneath. Remove the nut from the bolt using a crescent wrench. Lift up on the agitator to remove it. Replace the seals at the bottom of the agitator if they're faulty.
Removing the Tub
Remove the tub ring clips from the top of the inner tub. Beneath the agitator is the drive shaft. You'll need to remove this if the inner basket is shaking. Place a spanner nut over the drive shaft and tap it with a hammer to remove the drive shaft. Lift up on the inner tub to remove it.
Seals
Faulty seals on the outer tub can cause shaking, as well as a faulty drive block. Hit the drive block on the drive shaft with a hammer to remove but do not bend the drive shaft. Remove the nuts from the top of the seals at the bottom of the tub and replace them. Reassemble the entire unit after replacing the drive block and test.