We study optimal equity infusions into a financial network prone to the risk of contagious failures, which may be due to insolvency or to bank runs by short term creditors. Bank runs can be triggered by failures of connected banks. Under complete information on interbank linkages, we show that the problem reduces to a combinatorial optimization problem. Subject to budget constraints, the government chooses the set of minimal cost whose survival induces the maximum network stability. Our results demonstrate that the optimal equity infusion might significantly mitigate failure contagion risk and stabilize the system. In the case of partial information on the network, the controllers' focus swiftly changes from preventing insolvencies to preventing runs by short term creditors.