It would be a definite factor in open primary states. Probably no good data to show just how large a factor in any given election. But in states where there is little or no chance of electing a Dim to statewide office, why wouldn't many Dims take the opportunity to vote for the most liberal Republican candidate?
That definitely happened in Mississippi, even though it was in the general election rather than a primary. Plenty of Dims around who'd enjoy making mischief in a Republican primary.
Correction: the crossover voting by Dims in Mississippi was in the Republican primary runoff (recruited by Mississippi Rinos).