Anti-foreign and anti-Jewish sentiment being taken into account (although those are the asserted reasons for the quotas, it's not clear that there was any particular animus against Frank), I still think it would be a little difficult for the US to entertain taking mass immigration from Europe after the war had begun. There would be serious espionage issues.
I'm also curious as to why Otto Frank worked for 2 years in the US and left. At the time, it was a fairly long journey and it would seem to be a waste to just stay for 2 years.
Certainly there was nothing directed specifically at the Franks. The article describes his time in the US as an intern, at Macy's before WWI. I assume it was study or work related.