There is a ton of hiking around the area as well, so you can pick one of the many trails to explore with your dog when they are done playing at the beach and splashing around in the water. Dogs are welcome in any area of the recreation space, but they must wear a leash at all times - even when they are in the lake swimming. There is a lot of wildlife in the area and many of the species are considered endangered. Keeping these species safe from dogs is important to preserving the area.
You will not find showers or bathrooms near the beaches at Walter Lake, but there are vault toilets, drinking water, and showers up at the campsite in the recreation area. Always pick up after your dog when they are finished doing their business. Poop bags are likely not provided near the beach areas, so making sure you have your own will make it much easier to clean up after your dog. It also tends to get quite hot during the summer in the desert. Bring cold water and a drinking vessel for your dog so they can stay hydrated while out on the beach with you.
Overall, Walter Lake is a super fun and unique place to take your dog to the beach. Nevada has limited places to take your dog swimming, so this may just be the ideal spot for you and your dog!