A Los Angeles Times-USC poll recently asked Californians if they prefer a closed primary, a semi-closed primary, or the current top-two system. 50% support the top-two system; 40% support a semi-closed primary; 10% support a closed primary. See this story. The poll was conducted on-line between April 18 and May 18, before much of the publicity about how Democratic candidates for Governor and for U.S. House in certain districts are using insincere campaign messages to influence Republican voters to choose one Republican over another Republican.
Unfortunately the poll did not ask voters about a blanket primary. In a blanket primary, all party candidates appear on a single primary ballot, and all voters use that ballot. But the top vote-getter from each party then goes to the November ballot. Independent candidates do not participate in the primary, but petition directly onto the November ballot. California used that system in 1998 and 2000. The U.S. Supreme Court said it violated freedom of association for political parties to be forced to let outsiders help choose their nominees. But the system could be made constitutional if it were voluntary, and parties that didn’t want to use a blanket primary could instead nominate by convention.