曹老师的解答非常好！本题答案只能选 but。

本题如果要选 when，则从句需用否定式，比较下面的同义句：

Hardly a day goes by but I write to my family.

=Hardly a day goes by when I don't write to my family.

=Hardly a day goes by that I don't write to my family.

=Hardly a day goes by without me / my writing to my family.

我几乎没有一天不给家里写信。

注：我的解答仅作补充参考，请网友选择曹老师的解答。

Hardly a day goes by but I write to my family.

参考译文：几乎没有一天我没有给我家人写信。

A. when          B. since           C. but            D. before

D为正确答案but=that/which not

but可被看作关系代词，引导定语从句，在句中作主语，在意义上相当于 who not或that not，即用在否定词或具有否定意义的词后，构成双重否定。 

There is no mother but loves her children．

=There is no mother who doesn't love her children

没有不爱自己孩子的母亲。 

There was no one present but knew the story already．

=There was no one present who didn't know the story already

在场的人都知道这个故事。