Remove excess fat from meat. Cut meat into 1-inch pieces. Using a 3 1/2- or 4-quart slow cooker, place green beans and onion in cooker. Add meat. Combine in a medium bowl, salsa, beef broth, basil, and garlic. Pour over mixture in cooker.
Cover and cook on low-heat setting for 8 to 10 hours or on high-heat setting for 4 to 5 hours. Sprinkle each serving with cheese (optional).