First, see the Sears parts site for your washer:
See the Sears site sub-components for the agitator, basket and tub. It's possible that one of the gaskets or seals has worn over time.
I'd also pull off the cabinet and check the hoses leading to and from the pump. It's also possible that the pump is leaking.
See the following for how to remove the cabinet.
The illustrations also show how to remove the transmission.
The black oil / fluid that you see is likely coming from the transmission.
Since it seems to be "spraying" around the inside of the cabinet, I'd assume that it is leaking from the top seal between the transmission and clutch.
See the following for a parts breakdown of Whirlpool top load direct drive washers.
If you look at page 15, you'll see a Seal, Gearcase Cover. It is possible that that is the cause of your transmission leak.
The rust on the bottom of the transmission may (hopefully) be simply caused by the water leak.
I'd remove the gearcase cover (8 bolts) in order to check the gearcase cover seal, and to check the oil level. The Whirlpool transmission takes about 15oz or 80-90 weight oil.