Nowhere in the Constitution does it state that there is any professional requirements to serving in government. The only limitations are related to age and citizenship. There is simply no reason to create an rule stating that actors can not serve their country.
Yes, Arnold was a terrible governor, but there have been PLENTY of terrible governors who were lawyers and nobody is suggesting that lawyers can't serve in government. Banning actors from elected office would be unconstitutional and unnecessary. If you don't want to elect an actor, then vote for the other guy.
Every citizen has the right to hold government positions, and run for elections. This is without exception, regardless of ones social status. While there may be a popularity held by an actor or actress, generally, this is not viewed favorably by the voting public, as these people are not in politics.