Question

I asked the people how they were doing对吗？

Answer 1

不对。

do是及物动词，需要宾语，而how是副词，不能作宾语，只能把how变成what，what可以作代词，就能作宾语了。即：

I asked the people what they were doing. 我问人们他们在做什么

或者：

I asked the people how they were doing it. 我问人们他们是怎样做那件事的。

Answer 2

how 改为 what