Well, Republicans did participate in open primary until 2010, and then they stopped. I wonder if there's any specific reasoning I'm not aware of.
If someone searches enough, there might be a reason/statement in CA too. Anyhow, here's the one for Idaho, which did it at about the same time (2012):
You have to go back five years ago, to June of 2007 to understand how it all began. That’s when the Idaho Republican Party State Central Committee decided to close its primary to only registered Republicans. Jonathan Parker is the Executive Director of the state’s Republican Party. “We do believe that it is our right to essentially let Republicans chose Republican candidates, Democrats choose Democrat candidates, as these are the candidates who will be our standard bearers, carrying the torch for the Republican Party in November.”
TLDR: the official reason is that's how freedom of association should work. (And they won in court, as that story details.) FWTW
Boise State Political Science Professor Gary Moncrief [...] says on a national scale this isn’t unusual - for a faction of either political party to push for a closed primary. “The party that leads that charge is often, in fact almost always, the majority party in the state,” says Moncrief. “In some states it’s the Democratic Party and the faction within the party that tends to lead that charge tends to be the more liberal faction. And in the Republican Party, if they’re the ones leading the charge, it tends to be the more conservative faction.”
I suppose that does make it a bit more unusual for CA Republicans, given that a GOP majority there isn't a fact of life there.