Party leaders and donors fear that nominating either man would have negative ramifications for the GOP ticket up and down the ballot.
The meaning is
Party leaders and donors fear that nominating either man would have negative ramifications for the GOP ticket throughout the spectrum of elected offices.
Since in the US several elections are held simultaneously, the voter has to mark candidates in several positions in the ballot that he receives in the voting station. An excerpt from Wikipedia's entry on US elections slated for 2012:
The presidential election will take place alongside elections to the Senate and the House of Representatives. Several states will also hold gubernatorial and state legislative elections.
The offices range from high positions, such as that of the President, down to lower positions, such as the position of a US senator, and possibly even lower.
So an unpopular step by a party might disaffect the voters to such an extent that they will avoid choosing that party's candidates in all the positions on the ballot ("up and down the ballot").
From the (a?) grammar standpoint, "up and down the ballot" is an adverbial phrase.