This is because, since the advent of the technology that led to droids, they have been perceived as threats by biological beings throughout the galaxy. And the various revolutions and insurrections led by droids through the ages didn't help, nor did several incidents where the use of droids made things worst even if the incident in itself wasn't their fault.
Droid technology predates spaceflight technology in the galaxy, but it evolved with it as both technologies required advances in electronics and software.
According to Cynabar's Fantastic Technology: Droids, an RPG supplement by WEG:
As electronics and software compression technology improved, programming a droid with multiple skills became standard practice. Droid technology expanded and units were seen in systems across the Old Republic.
Resentment directed at these units appeared almost immediately. It was widely felt that the use of mechanical intelligence was the cause for massive job loss throughout much of the galaxy.
Then the technology evolved enough to give the droids an 'artificial sentience' which had it's advantages and it's drawbacks. It also meant that such droids could break the law. And they did, leading to many systems and organization, up to the Old Republic and the Empire, passing laws that are known as the "Droid Statutes" in the Imperial Code of Law.
Then there were the 'Great Droid Revolutions' starting with the most known that happened in 4000 Before Yavin on Coruscant, as depicted in The New Essential Guide to Droids. Droids went 'rogue', causing massive destruction and loss of life among the population, furthering the resentment.
The use of combat droids by the Confederacy and the existence of assassins droids didn't help to paint a nice picture of droids in general in the public's eyes.
Finally, what drove the nail in the coffin from a military standpoint was the disappearance of the Katan Fleet, described in The Thrawn Trilogy. Built in the waning days of the Old Republic, it was supposed to be the jewel of the Republic Fleet.
The whole fleet went a tour of the Republic, showing itself in what was a big PR operation aimed at it's citizens as well as a warning sent to the enemies of the Republic. The fleet was largely automated, with most functions performed by computers and droids, allowing for a very small crew when compared to other ships of the time. All ships were slaved to the flagship, which would be their undoing.
While the exact circumstances are not known, it seems that the crew became infected by some sort of virus during one of their stops, included the Command crew. Rendered delirious by the fever, the commanding officer ordered a blind hyperspace jump from the deck of the Katana, the flagship, and the whole fleet followed the insane order, vanishing to parts unknown until decades later.
(I'll try and add citations later once I can find them in the trilogy, it's huge and I haven't read it in a while.)