IMO, the same motives were at work for both alcohol and drug prohibition. This all came about during the Progressive Era. It was the time that gave us the 16th and 17th Amendments, as well as the Federal Reserve.
I'm sure there were a variety of reasons why people supported these efforts. Some were no doubt well-motivated reformers, some were meddlesome nanny-staters, and others, such as those in government, wanted a bigger government.
This is my speculation only - it seems reasonable that the hundreds of thousands of battle hardened young men, suddenly turned loose after a devastating war, could create problems. There was probably a fair amount of crime committed by veterans addicted to alcohol or drugs, but I have no numbers.
Since prohibition was a new concept in America, it was not unreasonable for the average citizen to think that if you outlawed drugs or alcohol, crime and other pathologies could be controlled.
The biggest thing to remember is that this (1900) is basically 50 years from the invention of the hypodermic needle. Cocaine was basically newly known (c. 1880) along with heroin. While Morphine had been around for about a hundred years, it had only been injected for about fifty years, and really used extensively by injection since the Civil War (c. 1865).
Combine this with the realization that addiction was compulsive, and beyond the will of the addict - a concept barely being entertained publicly by scientists (c. 1880), and one has a developing "perfect storm" against the use of both narcotics and alcohol, which, IMHO, lead to the legislative attempts.
I am not as quick to accuse as you might be, nor am I as wiling to distrust these folks, with 47 states passing laws along with the federal laws as well.
I would grant you there may well be some proto-progressive behavior here, but this is somewhat in the future still. The "New Deal", and the Great Depression are some years away.
The trend I see, spanning 1800-1865/1865-1900/1900-14 is a whole nation caught unaware by it's own devices, a slow discovery of a real and disabling problem, and a somewhat knee-jerk, but well intended attempt to fix it.
The only part of this that I see as unlawful is the Harrison act, and perhaps the Pure Foods Act as well. Both of these were prototypes which later drove unlawful federal power. These were the first bites of a bitter fruit, as it were.
Surely it is within the states' rights to ban or regulate substances, and in good part, those bans and regulations were in place prior to the federal legislation.
Would you admit that it is within the rights and prudence of the states respectively, or do you seek a federal remedy, legalizing all substances nation wide?