If you’re seeking out a beach near Monterey, I suggest Asilomar State Beach in Pacific Grove just outside Monterey. The beach is narrow and stretches for about 1 mile down the coast.
If you’re a runner or just enjoy a nice walk, the walking trail/boardwalk is a nice place to get some exercise and see the coastline. From here you can head into Pebble Beach and walk along the outer edge of the famous golf courses. Along the road there is a bike trail that takes you along the beach. Dogs are allowed, but are supposed to be on beaches (though many rarely are). Overnight lodging is available nearby with great views of the surf.