I'm going to take this bit by bit and do what I can to highlight major opponents. Keep in mind there were a lot of groups at play here, and it was a very complex and took a lot of time to get the amendment to the US Constitution passed. If you want some basic history on the voting of women before/after the 19th Amendment passed, you can read my write-up here. Part of it concerns the electoral outcomes of the elections prior and after the passage of the amendment, but another part of it concerns women's voting patterns and how they were able (or unable) to vote prior to the amendment's passing.
I did manage to find many discussions of the opponents of women's suffrage throughout my research into the subject (of my own, to be clear, I am not a professor on the subject). The most commonly cited example is the The New York State Association Opposed to Woman Suffrage, but they are fairly easy to find information about and their name just about says it all; they're similar to the other organizations that felt the same about women's suffrage. So to start, let's take a look at a piece of history I found:
Veterans of voluntary service, Dolly Blount Lamar, Mildred Lewis Rutherford, Rebecca Latimer Felton, Mary McClendon, and Ida Cheatham politely greeted each other on the grounds of Georgia's state capitol late on the afternoon of July 7, 1914. Wearing their Sunday best and clutching the well-worn pages of their carefully rehearsed speeches, they climbed the steps of the gold-domed building and walked into the political sanctuary of the Georgia legislature. On any other day, these white Georgia women might have been cooperating on a campaign to prohibit the sale of alcohol, to raise money for a Confederate memorial, to protest the content of high school history texts, or to lobby for some "municipal housekeeping" issue such as public kindergartens. This time, however, they were talking to the House Constitutional Amendment Committee about woman suffrage, and they vehemently disagreed with each other. Lamar and Rutherford denounced suffrage in two lengthy speeches. Felton, McClendon, and Cheatham followed, arguing that woman suffrage was vital to the South 's future. The legislators defeated the amendment by one vote and recommended that the House reject the suffrage amendment. On that hot summer day the anti-suffragists won the first of many battles that would, seven years later, culminate in Georgia being the first state to vote against the ratification of the Nineteenth Amendment.^^1
So yes, there were women against it; many of these women were prominent and active in Georgia, and had connections to the Democratic party through family or marriage. This, however, is just one example of the narrative of suffrage, and in the South (this author especially) this was seen as a vote to decide how patriarchy would survive in the South. Many feared a precedent, as well, that would eventually allow black women to vote, as one example. The weakening of the state's right to choose who voted, and the challenge to white male supremacy in voting, was a huge deal in Georgia.
In 1911, California voted on the women's suffrage amendment called Prop. 4 (numbers repeat, so if you've seen this number for another initiative then it's not wrong, but this was a Senate-proposed amendment the people voted on). The 6th state to approve women's suffrage, it didn't pass without opposition. Anti-suffragists were there, arguing against it. Their arguments, also, were different from some of those in Georgia, and appealed differently to anti-suffragists. "To give woman the vote, said some passionate opponents, would pull her from the pedestal where she belonged, rob her of femininity, endanger her health, weaken her character, and lead to the disintegration of society..."^^ 2
While the arguments were many and varied, both parties were very lukewarm regarding support for the idea. Only the Progressive Party, which Teddy Roosevelt championed in 1912, had mentioned support for women's suffrage plainly and (if memory serves) in their platform as well. Wilson only began outright-supporting suffrage, according to this, around 1918. That in and of itself suggests the very lukewarm support that the idea received; there was not much urgency prior to get this done, and only after the protests had gotten increasingly rowdy (culminating in arrests of women who had been picketing outside the White House as President Wilson passed in and out) did the impetus for action really start to hit. Even then, the passage of the 19th Amendment was not swift, even if it was relatively bipartisan. Delaware, Maryland, Virginia, Alabama, South Carolina, Georgia, Louisiana, and Mississippi all rejected the 19th Amendment initially. The southern states only ended up ratifying it in the 50's or later, a full 30+ years after it was added to the Constitution. Now, if you want to go by how those states voted in the 1920 Presidential election, you'll find that most of the states opposed to it were Democrat-voting in the 1920 election, even though Wilson did eventually support the movement prior.
On January 12th, 1915, the New York Times reported on the failure in the House of Representatives to pass a women's suffrage amendment. As it points out, Wilson had said recently that women should fight for this right in the states, not Congress (a similar way that gay rights are being discussed now, and similar to how Douglass proposed the question of slavery be solved prior to the Civil War, it's a common theme).
In case you were looking for the vote count, here it is (keep in mind the 174-204 vote was overwhelming as a loss because the amendment needed a two-thirds vote to pass, not a simple majority):
Voting for the resolution were 86 Democrats, 72 Republicans, 12 Progressives, 3 Progressive-Republicans and one independent, and against it 171 Democrats and 33 Republicans.
That comes from the same article. So if you wanted to call it a partisan divide, the Republicans were "more" in favor and Democrats more" against, but there was really less of a partisan divide in terms of party and more of a divide on basic opinions within the parties.
On March 19th, 1914, the Senate had voted on a women's suffrage amendment as well. It was 35-34 in favor, but required that two-thirds majority, which it fell far short of, meaning it still failed. I don't recall who voted how, but that proves that the movement really gained steam within a few years to get passage. In 1878, the Senate had also voted on this amendment, and lost 16-34. So in the roughly 35 years between 1878 and 1914, the measure barely gained a simple majority, and lost by 11 votes. By 1918, the measure lost by 2 votes. In 1919, by 1 vote.
As for prominent women against it, there were some, as I listed in the Georgia case. The National Association Opposed to Women's Suffrage, which was founded in New York in 1911 (not to be confused with the New York State Association of similar name that was founded earlier), was founded by a woman named Josephine Dodge.
Many women rose to prominence on both sides of the aisle, but we both know who won, and the fight (complex, long-fought, and a hard battle all around) ended as states began to ratify suffrage independently (9 states gave women the right to vote by 1912, and Illinois allowed women to vote for the presidency in 1913) and as the movement gained continued increases in support from prominent figures.
Hope that answers your question!
^^1 Caretakers of Southern Civilization: Georgia Women and the Anti-Suffrage Campaign, 1914-1920 Elizabeth Gillespie McRae The Georgia Historical Quarterly, Vol. 82, No. 4, GEORGIA WOMEN: PERSPECTIVES ON CLASS, RACE, AND ETHNICITY (WINTER 1998), pp. 801-828
^^2 Why Women Should Not Have the Vote: Anti-Suffrage Views in the Southland in 1911 Jane Apostol Southern California Quarterly, Vol. 70, No. 1 (SPRING 1988), pp. 29-42 Published by: University of California Press on behalf of the Historical Society of Southern California