The civil war first started after they had conducted their anti-democratic coup, though.
They got about 10% of the votes in the elections for the constituent assembly due to write the new constitution. They responded by overthrowing the Duma and calling off the assembly.
I agree with you that many of their members probably did support democracy - in fact, many of them lost their lives for holding on to that belief and resisting an increasing authoritarian leadership.
I didn't make the distinction mainly for simplicity because even if we draw a clear line and go all black and white and call every Bolshevik anti-democratic, the majority of socialist and communists in Russia were not Bolsheviks.
Part of the problem with the Bolsheviks, to me, stands with Lenins idea of the vanguard party, which was part of the issues surrounding the disagreement between Martov and Lenin that led to the split: With the idea that they made up a revolutionary vanguard, coupled with the fact that they were outlaws for a large part of their existence, they conditioned themselves to believe they "knew better" and that they would have support, and that anything indicating anything else was just another part of government or counter-revolutionary propaganda. Then with the creation of the Soviets following the February revolutons - standing against the Duma as an alternative seat of power -, they surrounded themselves with people who supported them, and this helped them legitimise to themselves the idea that they did actually have support.
When the constituent assembly elections provided evidence that showed they had pretty much no support outside of the big cities, it was easy for them to dismiss it as the result of "counter-revolutionary" activity etc.
The civil war certainly made it worse. It both provided good reason and an excuse to crack down on opposition. When parts of the opposition chose to fight on the other side because they had legitimate reasons to fear a party that had just conducted a coup, it just made it easier to crush them and write them off as traitors and counter-revolutionaries.