With its military history, there are a number of museums and naval sites that can be toured, but there isn't much for dining or nightlife. Nearby Treasure Island, which is actually a manmade island, has a few restaurants, wineries, and a coffee shop and is only a 5-minute drive from Yerba Buena.
There are bus lines that run between the two islands as well as into San Francisco's core. The distance between Yerba Buena Island and San Francisco is drivable. On days without traffic, it takes as little as 10 minutes.
Although many roads are closed off, there's plenty of space to walk around this moderately sized island which, between it and Treasure Island, is a collective 580 acres. It's not likely that you and Fido will encounter as many pedestrians as you will in San Fransisco's city center. It can be quite foggy on the islands as well as extremely, at times dangerously, windy as the breeze whips up off of the Pacific Ocean's waves, so dress accordingly if you do plan to explore the island with your pup.
Blue Park is the largest park on Yerba Buena Island and so it tends to be the most visited. The area tends to feel like a tranquil escape from city life with amazing views of the ocean and the architecturally stunning Bay Bridge.
Directly off of Treasure Island Road on the spine connecting the two islands is Clipper Cove Beach. Locals love this spot as it's quaint and is partially protected from the harsh eastward winds and boasts some of the most gorgeous views. If your dog is well-trained off their leash and loves water, then this may be the best spot for you on Yerba Buena Island.