My kenmore 400 top load washer will not run the cold water cycle at all. It will also stop on the rinse cycle because the rinse cycle is cold water only. What could the problem be?

check screens in the water inlet valves they do plug up with dirt

if you're saying it dies in the rinse cycle you could have a problem with your cold water not coming into the washer,the washer only uses cold water in rinse cycle,no cold water the washer won't run,try starting it just with cold water and see if it fills,if not bad water valve or problem with water in house valve or bad hose.

When the cycle doesn't advance, it's probably the timer or a cold-water supply problem:

It's the timer, if your washing machine fills with water and begins agitating, but the timer never advances--or if the washer is in a spin cycle and the timer won't advance. Then you need to replace the timer.

It may be a cold-water supply problem, if the washing machine fills with water, agitates, drains, and spins, but then doesn't fill with rinse water. See There's no cold water.

If no water entered the machine at the rinse cycle, then I would check for a cold water supply problem, since cold water is what is used in the rinse. Check the water inlet valve. Remove the water supply hoses and check the small screens inside the water inlet valve. Make sure they are clear of rust or debris. Pay particular attention to the cold water side. Next, check the solenoids at the water inlet valve. They can be tested using a multimeter or ohmmeter. Check for continuity. A lack of continuity indicates a bad solenoid. I will attach a photo for reference. Check this and if you need additional assistance let me know. I am working with limited info. For instance, can you advance the cycle manually and make the machine operate? Sorry for your trouble and I hope this helped. Best wishes.

Are you on well water? turn off the cold water to the washer and clean the screen filter out. do not leave filter out.
the machine probably has a cold only final rinse so it won't work properly.

When the washer initially fills for the wash, is it only hot water entering the drum? Try switching the water temperature between hot and cold as it fills. If any of them show little or no water getting through, it sounds too easy but check to make sure water supply taps are on, and that the cold/hot fill hoses aren't reversed. Remove the fill hoses and check any of the screen filters for blockage (in the hoses and in the inlet water valve). When the machine stops and waits to fill but all you hear is a hum, it sounds like there is power going to the valve but water is not able to get through it.
Good luck!

most likely cuase would be the inlet valve
you would need the complete model number to make sure you get the right part for that unit.

If you are getting no water spray during the rinse cycle that would cause the soapy clothes issue.

Are you getting cold water during the wash/rinse cycles? If you have the machine set to warm/warm you may only be getting hot water while the final spin is calling for cold.

If you have cold water available then the timer is not calling for the water spray. The problem is either in the mixing valve or the timer.

the timer change by itself. You don't have to change anything.
But cold rinse only works if you have the cold water connect to the washer

POWERUP, YOUR WASHER COULD BE THE LID SWITCH OR THE COLD WATER VALVE. AT THE END OF THE WASH CYCLE, THE MOTOR WILL SPIN OUT THE WATER, BUT ONLY IF THE LID SWITCH IS ACTUATED. IF THE SWITCH IS BAD, IT WILL NOT SPIN OUT THE WATER AND THE TIMER WILL NOT ADVANCE FROM THAT POINT.
IF THE WASHER SPINS, THE NEXT STEP IS TO FILL WITH WATER FOR THE RINSE CYCLE. MOST WASHERS USE COLD WATER FOR RINSES. IF THE COLD WATER VALVE IS RESTRICTED OR FAULTY, THE WASHER WON'T FILL. IF THE WASHER WON'T FILL, THE WATER LEVEL SWITCH WON'T ACTIVATE AND THE TIMER WON'T ADVANCE TO THE NEXT STEP.