I'm hoping cenus information and historical gun manufacturing records make this an answerable question. What percentage of the population in the Old West actually carried handguns? Was it a minority or a majority?
The answer differs between "carried" and "owned", and "in towns" vs "not".
I'll only cover the first of the 4 combinations.
You did not carry guns in many towns. Tombstone, AZ prohibited carrying firearms, as did Deadwood, as IIRC did Dodge City. The ordinances prohibited it, the signs indicated that you had to check in your firearms at the Sheriff's or special points.
As a matter of fact, the infamous fight at the OK Corrall was technically over the issue of police chief Virgil Earp and co trying to make the outlaws comply with the city law and disarm.
A great deal of detail on town in the Old West prohibiting carrying can be found in this Pink Flamingo blog post - while it's a blog and not a peer reviewed article, the author is somewhat of an expert on Wyatt Earp specifically and Old West in general, and is a pro-gun Republican, so it's hard to read some political subtext/spin into the info.