If you're looking to just clean it then soapy water or tooth paste and a nylon brush will work just fine. If you want to get some of the luster back to the brass then try some polishing compound and a brass brush. I use the same stropping compound paste that I use to polish and hone my knife blades and it works well. Just smear some polishing paste on it with a cloth and gently work on it with the brass brush. Rinse and repeat until you get the desired effect. Buff it with a clean cloth towel when you're done.
It does remove some metal so I wouldn't do this on a rare or valuable relic but for everyday finds like this brass badge it's fine. It will bring out some of the details and make your brass finds look a lot more interesting.
You can get the brass brushes and honing compound on Amazon fairly cheap.
Brass Brushes:
https://www.amazon.com/dp/B08DM15R4V/ref=dp_prsubs_2
Honing compound (6000 to 8000 grit):
https://www.amazon.com/Compound-Tor...8988&s=hi&sprefix=honing+com,tools,153&sr=1-4