I would be very curious to understand what aspects of the government shutdown would be impacting putting on such a game overseas. Exactly what costs would rely on appropriated funds in order to pull this off: perhaps additional costs for extra security and for processing base access to an influx of civilians?
Morale, Welfare and Recreation (MWR) activities (gyms, fitness centers, ball fields, swimming pools, golf courses, base theaters, etc.) are funded through Non-Appropriated Fund instrumentalities, that operate outside of annual appropriations passed by Congress. Other military base activities and support services, such as commissaries (food) and exchanges (department stores) as well as some on-base lodging accommodations, also operate on NAFs.
Unlike appropriated fund activities, MWR NAFs can be augmented by additional funding avenues, such as proceeds from air shows (Ramstein holds a large air show every year) and certain fund raisers (casino nights, antique auctions, etc.). Truth be told, when I first heard about this game, I figured ESPN was probably paying something for the broadcast rights and thereby help contribute to MWR revenue.
Hopefully the base commander and comptroller folks at Ramstein can sort out the details quickly.