My Kenmore 70 Seriies Heavy Duty Super Capacity Plus wash machine's cold water cycles stopped working today. The hot water cycle (only) works, but any cold water cycle does not work. The cold water line is open. What is likely wrong and how do I fix it?

You have to do a little research here, to check what is going on in your washer :
Most probably you require a multimeter or a volt meter.

First close the water tap, take off the water hose at the side of the washer, and insert in a bucket. Open the tap shortly to see water is coming out, and to clear possible dirt in the hose.
Look in the screw connector to which the water hose was connected : anything obstructing the water flow there?
Reconnect the water hose and open the water tap.

Remove the top or rear cover, and have a general look as to what is going on.
Now check the water valves inside the machine.
Switch the power on and start the washer.

Outside the machine you have the screw coupling, directly connected to an electric valve (called a solenoid) at the inside of the machine.
Depending on the program one of the two valves is opened to partly fill the tub (*1*).

There are two possibilities : either the cold water valve is faulty or the electric signal that opens the valve is absent.

To check what is wrong, select AC - high Voltage (250 or 400 Volt setting) on a multimeter. Now you can measure if there is an electric signal on the two connections of the valve.
If it is there, the valve has to be replaced (available at most washing machine repair shops).
Often you also can hear if the valve is activated. You should hear a clear humming.
If there is no electric signal, do one last check or ask a professional repairman to have a look.
The last check refers to action (*1*) : the partly filling of the tub : this is a protection to prevent the entire tub to fill and overflow.
This protection is provided by a pressure switch : it is a flat rounded unit with a 1/2" rubber hose connected to the lower side of the drum.
When water is entering the tub, the water level will rise. Air trapped in the rubber hose is pressurized by this, and the higher the water level gets, the higher the pressure becomes.
At a preset maximum level the pressure switch closes the water valve.
Find the water pressure switch and check if the rubber hose is blocked by something. Take the rubber hose off the tub, and blow on it : you should be able to hear the switch clicking.
If not, the pressure switch may be faulty. Note the wiring down on paper, remove it, and have it checked at a repair shop. If faulty replace it.

Of course you can also come back to the board and discuss your findings!

Success !

:)

If the hot water is working, then the pressure hose and level switch is OK. Electrical causes for no cold water include valve, wiring, connector, temperature selector, and timer.