As far as I know, at the beginning of the war we knew that the Nazis were interning enemies of the Reich. Many of these people were later discovered to have been worked or starved to death, bit at this time it does not seem that the Nazis yet had camps built for the specifc purpose of killing massive numbers of people; that seems not to have begun until we were already in the war.
While the Nazi use of the concentration camps was particularly awful, similar types of facilities had been used by other nations, for example, our internment camps during WWII for Japanese-Americans, England during the Boer War, and Cuba.
The issue of the camps is a tricky one. It seems that the Allied governments were aware of rumors but had no hard evidence of the extent of what happened in the camps until 1942, at which point everyone was in the war.
Overall, it is difficult to tell how much the governments knew, but however much or little was known, it was not at all part of the reason for anyone’s entry, no. The invasions and alliances were sufficient under international law to justify their entry, altho some have argued that the US entry in Europe was a bit of a stretch.