I find it very doubtful that Blizzard will include an option to block all Starter accounts from playing an Arcade game. They wouldn't have created Spawning if they wanted that to be possible.
If on the other hand, Blizzard would take it upon themselves to ban players who hack Mafia games (and Arcade in general), allowing in Starter accounts wouldn't be so much of a problem.
(Ideally they would also provide an option to disable lag screen chat, removing the need for Mafia staff to ban players for in-game cheating.)