I have a leak between the the polybutylene pipe nut ring cone and the valve for the cold water on the washer. What do I need to fix this? The leak is small and is just dripping.

 

If it's gray then it's Quest and should all be replaced with PEX.
Google "Quest recall".
Box is all distorted, old style gate valves that always leak, if it was mine I'd replace the whole thing including new hoses to the washers.
SharkBite 1/2 in. Washing Machine Outlet Box with Water Hammer Arrestors-25031 - The Home Depot
Those old rubber lines can burst, us braided lines instead.

 

If this is in a mobile home, you may have Quest piping throughout. It is a problem product and should be replaced, but at a considerable expense. While much of the problem came with the connections rather than the pipe itself, you can buy Polybutylene to PEX connectors and crimp them to the PB. It isn't a fix all but will make a better connection at the outlet box Joe mentioned than what you have. The one problem will be buying the crimping tool, which can run up to $65.

You can use the PB to PEX Sharkbite as pictured here SharkBite 1/2 in. Brass Push-to-Connect Polybutylene Conversion Coupling-U4008LFA - The Home Depot. The gray end goes to the PB and the other end goes to the PEX.

Short pieces of PEX are inexpensive.