We have this situation in California.
The primary is only for the party, it has nothing to do with rights or freedom.
The parties should be able to choose their own candidates, they should not be open to the opposition party choosing their candidate for them.
In effect a opposition party queering the other sides primary has negated the actual official governmental election.
In Florida for instance the Nelson democrats could literally have chosen his Republican challenger, they could have even had a sympathizer run in the Republican primary and chosen him.
(Please no remarks about the Florida primary)
A few decades ago, the Democratic party in Southern states was whites-only, claiming that as a private organization it was free to discriminate. Because the Republicans weren't competitive in those days, and the Democratic primary effectively decided the general election, that helped to ensure that if any blacks managed to register and vote, their votes wouldn't mean much.
I don't see how letting party bosses who weren't elected or even appointed by elected officials tell the voters which candidates they may choose from would benefit democracy. The hypothetical abuses of open primaries pale in comparison to the actual abuses that occurred in closed primaries; crossover voting might swing a close race, as happened to McKinney in 2002 (I don't think it was that close this time out), but I don't know of any instances of wholesale hijacking of one party's primary by the other.
In your hypothetical, if the Democrats were so dominant that they could swing the Republican primary their way, then wouldn't they be dominant enough that it wouldn't matter who the Republican nominee is?