step up one's 〜 against ...

「～を阻止するために...に力を入れる、強化させる」。

...には、battle 、effort、movesといった表現が入って、あることに対して、それを阻止するために、働き、動きを強化させる、というような意味で使います。

BBCの記事、'Floods affect 20m people - Pakistan PM Gilani'から。

'Pakistan's PM Yusuf Raza Gilani says 20 million people have been affected by the country's floods, a much higher estimate than the UN's 14 million.

He was addressing the nation during much muted celebrations of Pakistan's independence from the UK 63 years ago.

The United Nations has confirmed at least one case of cholera among the victims, in the Swat valley.

Health workers have been stepping up their battle against waterborne diseases in the flood-hit country.

In his address, Mr Gilani said 20 million were now homeless, although it was unclear how many of those people were briefly forced to leave their homes and how many had lost their houses altogether.

"Unfortunately, the recent unprecedented torrential rains and devastating floods have made more than 20m people homeless, destroyed standing crops and food... worth billions of dollars, washed away bridges, roads, communication and energy networks," he said'.