1. To stop the flow of something: The doctor said that it was important to staunch the ruptured blood vessel from bleeding.

The authorities mentioned that the only way to curb or to staunch the flood of illegal drugs into the country was to have check points stationed at all airports and at all the roads at the boarders.
2. To lessen the origin or cause of a leakage: Timothy had to staunch the gush of water by closing off the main valve.