In this case:
Down = Lower
Ballot = The US voting system where we record our votes on a piece of paper (also known as a ballot)
US government is organized at Federal, State, County, City, Town etc. levels. Colloquially, levels of government go from high (federal) to low (city, town). Down ballot Republicans refer to Republican candidates running for lower levels of office. In this context, if you're reading about the Presidential race (seen as the nation's highest office), down ballot Republicans would refer to everyone running for offices below that of President.