Both staff and customers will also need reassuring that the company has sufficient financial resources to survive the economic downturn.

本句有两个结构需要弄清楚：

（1）need doing = need to be done    

need reassuring = need to be reassured

（2）reassure sb.（间接宾语） that... （直接宾语）    被动式为：

sb. is reassured that...

两个结构结合为：

sb. needs reassuring that ... 等于：

sb. needs to be reassured that ...

reassuring 是动名词，作 need 的宾语。而 that 从句是 reassuring 的直接宾语，Both staff and customers 是 reassuring  逻辑上的间接宾语。

所以：

Both staff and customers will also need reassuring that the company has sufficient financial resources to survive the economic downturn.

Both staff and customers will also need to be reassured that the company has sufficient financial resources to survive the economic downturn.