If you followed the debate about California's Prop. 14 recently, you know the logic here. Basically, when you have a closed primary (that is, one that's limited just to party registrants), you end up with an ideologically extreme electorate that will only pick other extremists for office. If you open up the primaries to independents and moderates, however, you get less extreme officeholders. At least that's the idea. Does the evidence bear this out?
No, not really. Eric McGhee, Boris Shor, Nolan McCarty and I presented some research at APSA last week looking at roll call voting in all 99 state legislatures, breaking the states down by the rules governing primaries. We found no consistent or clear effect of primary rules on the voting behavior of elected officials. States with open primaries have legislatures that are just as polarized as those with closed primaries. You can read the paper here.
Why might this be the case? Well, one reason is that the insiders who dominate primary elections aren't really hampered by a primary electorate that includes moderates. As long as they can limit the group of candidates to just ideological extremists (which they can do through the manipulation of resources like endorsements, funding, and expertise), they can prevent moderates from getting nominated.