Because they were at war.
Anything that happened afterwards is totally irrelevant. You cant make a decision for then looking backwards, you need to put yourself in their position at that point in time.
They were essentially fighting the Rachni Wars 2.0. It doesnt matter how irresponsible, stupid, idiotic, and intrusive the Salarian's involvement with the Krogan were, the fact of the matter is now (as in, then) they're killing with no sign of stopping. Negotiations have failed. The Krogan have annexed from the Council. Theyve even dared the Council to take back their own worlds.
You can keep fighting, and people will continue to die on all sides, all the while youre losing. You can do nothing, accept defeat, and say goodbye to more lives while the Krogan over populate the galaxy. Or you can do something, anything to turn the battle in your favour.
The Krogans best weapon was their rapid breeding. No matter how strong the Turians, the Krogan bred quicker. Military forces were quickly replenished. So they attacked their breeding to stifle their war efforts.
Maybe it was managed poorly, maybe it lacked morals, maybe maybe maybe. But nobody here has come up with all that much of a better solution. Im not necessarily condoning the genophage, despite my posts, but I do believe its extremely naive to label it as evil and unnecessary without offering a different solution to what was an galactic problem.