I heard an argument recently that at the start of the suffrage movement, the majority of women did not want suffrage. The argument stated that the suffragettes were working more to convince women to vote in referendums than for lawmakers to vote for an Ammendment.
I'm no expert in this subject matter but I'll try to get the discussion started by providing what I've previously studied and background from internet-friendly sources.
Part of the understanding of women's suffrage must also take into account the state-by-state advances to granting women the right to vote. The areas of what are now Wyoming and Utah granted the right to vote to women well before the federal amendment or even national associations were formed: the Wyoming Territory as early as 1869, in the hopes of boosting their population size, and in the case of Utah, balance the non-Mormon voting pool.^1
Many leaders of the suffrage movement favored the state-based approach, led by Lucy Stone, whereas Susan B. Anthony and Elizabeth Cady Stanton initially favored the federal approach via Constitutional amendment. The federal government had a ruling in this matter in 1875:
In Minor v. Happersett (1875) the Supreme Court ruled that women could only receive the vote as a result of explicit legislation or constitutional amendment, rather than through interpretation of the implications of the Constitution. In a unanimous opinion, the Court observed that it was “too late” to claim the right of suffrage by implication. It also ruled that suffrage was a matter for the states, not the federal government, to decide.^2
The National American Woman Suffrage Association (NAWSA) was founded in 1890, and initially did not advocate for the vote on the principle of equal rights but instead that women would "uplift politics" and provide a counterbalance to the immigrant population's vote.^3
I have not been able to find indication that the overall suffrage movement, which in and of itself spans decades and had to compete with WWI in the beginning of the 20th century, was aimed mostly at convincing women to advocate for suffrage, but there were certainly women's organizations opposed to the vote. The suffragists often welcomed discussion with these groups, calling for open discussion and debate. ^4
Those typically opposed to suffrage were wives of politicians who did not want to their access to political power eroded by suffrage, or those that advocated that the women's natural place in society was as a home-maker and caretaker. By the 1870's those opposed to women's suffrage also organized and became more public and in 1911 the National Association Opposed to Woman Suffrage was formed.^5
TL;DR: The argument that the suffrage movement was mostly to convince women to advocate for the right to vote would appear to be an oversimplification of a decades-long struggle across varied political climates, but there were definitely women's groups formed against it.
^5 (p.32) http://books.google.com/books?id=d9lhBw8t410C&pg=PA32#v=onepage&q&f=false