Many think that for fear that sb. should do sth. is the use of subjunctive mood.
And they are right.
The
should/shall in the first person convention is appropriate for the ordinary future and conditional. Here we have a subjunctive created by a modal auxiliary and this would naturally be
should rather than
would where we wanted to give a more formal feel to the expression. One could also use
might.
I agree with Goldenband that one could use either
should or
would here, or one might elide the auxiliary altogether - which would give an even greater feel of formality.
Here are instances in the GB and US corpuses of the different forms.
For fear that he should fall behind - 0 (GB) - 0 (US)
For fear that he would fall behind - 1 (GB) - 14 (US)
For fear that he might fall behind - 2 (GB) - 7 (US)
For fear that he fall behind - 1 (GB) - 0 (US)
Lest anyone think that means that the form with
should isn't to be found, here are two examples from literature, one American, the other British.
And, for fear that he should suspect how her heart was aching, she gave a particularly brilliant and joyous smile - Eleanor H. Porter
» Miss Billy Married
» Chapter XX. Arkwright's Eyes are Opened.
Is there any one you want to see, or anything of that sort?' and doing it as steadily as possible for fear that he should mistake the carelessness of whisky for the distraction of fear, I got my candle alight - H.G. Wells
» The Story of the Inexperienced Ghost.
The Google Books ngrams (
click) illustrate the increasing currency of
would against
should in this construction, in both American and British English.