I suspect a lot of people opposed the 19th Amendment, and similarly opposed the 15th, because they felt that voting was something the states should individually determine, based on their own circumstances. They felt that the qualification to be a voter was not a power originally given to Congress by the Constitution, and it should stay that way.
At the time of the 19th Amendment, some states granted women the right to vote, some did not, some had lower voting ages than others, some did not, some had certain tests to vote (literacy test, ownership of property) and some did not. The argument would be that Congress was in no position to determine for each state exactly what the right qualifications for voting were -- that should be left up to the people of each state to define. And if, for example, the particular culture of Nebraska said women should vote, and of Alabama that they should not, then the right of the people in each state to make up their own mind should be respected by Congress.
The argument isn't very strong, once the 15th Amendment passed, of course (and that one passed largely because Republicans wanted to ensure dominance by acquiring the black vote). It is also philosophically weakened by the presence in the Constitution itself (Article 4, Section 4) of the power of Congress to ensure that each state had a republican form of government.
That is, the Constitution explicitly prohibits Alaskans, say, from voting themselves into a dictatorship -- voting to give the governor all power and dissolve the Alaskan legislature. If the Constitution gives Congress *that* kind of power, it's hard to see how, philosophically, it isn't at least somewhat the business of the United States to set nationwide standards for who can, and cannot, vote, since that is the basis of the "republican form of government."
But keep in mind the century between 1820 and 1920 was a steady evolution of the nation from one where *any* activity of the Federal government beyond what was strictly necessary -- what simply could not be accomplished at the state level -- was viewed with great suspicion, to a nation in which "federalism" is kind of a joke, where the power of the states is severely circumscribed by the power of the Federal government. Opposition to efforts to define the nature of the government of each state was natural and explains much of the opposition to the specific changes made during this period to bring much more of the life of Americans under the scrutiny and control of the national government.
As for the idea that the 19th Amendment was opposed by men not wanting to give up power to women on the grounds of sexism -- this is absurd. The 19th Amendment was proposed and passed entirely by men. Not a single woman voted on the issue. The ONLY reason it passed was because men, in general, felt morally obliged to share power with women. It is one of the single best examples of selfless non-sexism ever, and if there is a parallel historical case the other way -- a case in which women, having all the power, voluntarily surrendered much of it to men, simply because they felt it was the right thing to do -- I don't know of it.
I hope that this is the answer that you were looking for and it has helped you.
