What is the ruling of the Shari'ah on two brothers earning money through bribery, and how should one deal with their money and their relationships?

Bribery is one of the major sins. The ruling concerning your brothers' income depends on the nature of their work: if it is permissible, then the income is permissible; if it is forbidden, then the income is forbidden. Bribery has no bearing on the permissibility or impermissibility of their salary, provided they are performing their assigned work to the fullest extent.