When someone 'puts their stamp' on an idea, it means they agree with the decision or plan. The term 'stamp of approval' has been shortened to just 'put their stamp'. A stamp of approval could be verbal or written. I'm not Chinese, but I think a good comparison is a 'Jian Ding' stamp, that shows that someone/company is approving of the product and wants people to know that they believe the product is made the right way.
In the story, Hester, the woman, decided to close the hidden stairways, and everyone who has lived in the house agree with her decision to close the stairways (they put their stamp of approval/agreement related to her decision).