To ensure that fewer books were lost

[Nonverbis]

This is Upstream Proficiency by Virginia Evans and Jenny Dooley.

Could you tell mey why do they use "were lost"? I would say, "ensure that fewer books should be lost", or "ensure that fewer books be lost".

 

[teechar]

This is from Upstream Proficiency by Virginia Evans and Jenny Dooley.

Could you tell me why do they use "were lost"?

The past tense is often used in tentative situations (here we have the tentative "would") to distance the context from absolute certainty. In a similar manner, the use of will goes with the present and suggests a stronger (more likely) statement:
A new filing system will ensure that fewer books are lost.

I would say, "ensure that fewer books should be lost", or "ensure that fewer books be lost".

No. See above.