Yes, they do influence them and not infrequently, the one who wins the Globe and SAG especially wins the Oscar.
But other issues can creep in, like the potential that the sexual harassment/assault charges may have knocked James Franco out of the Oscar race this year. For those who don't know the "Norbit" effect, it's widely believed that Eddie Murphy was on his way to an Oscar for Best Supporting Actor in 2006. He won the Globe and SAG and other awards up until the Oscar for his amazing work in "Dreamgirls." And then before the Oscar voting, his movie "Norbit" was released to wide disdain and condemnation. It was a totally offensive movie where among multiple roles, he portrayed an enormous, black female she-devil (her name was "Rasputia," for cripes sake...). The movie was panned and he got nailed by the critics and public for that portrayal. On Oscar night he sat waiting for his name to be called, and then Alan Arkin won for "Little Miss Sunshine." Murphy supposedly exited the auditorium immediately afterward, and his career has frankly been on a downward slide since that night.
Next rule for the Oscar nominees: don't PO the voters...