They mostly believed that women would become political and devoted to causes and participating in politics which would lead them to abandon their duties towards their homes which was mostly about things like chores and being a housewife. This was the common belief among anti-suffragists, but there were also radicals who believed even worse things.
A grand jury is involved early in a case. It is up to them to determine whether or not charges should be brought against a suspect. A trial jury, on the other hand, is involved at the end of a case, when it goes to trial.