Primaries and voter suppression
I suspect the Sanders camp is wrong on the counterfactual—they would not have won in an open primary. For one thing, Clinton was always up in the pre-election polling by a hefty margin, which isn't what you'd see if there was a mass of voters intending to vote Sanders but then found out they couldn't. However, I find myself agreeing with Matt Breunig:
"The story is supposed to be that Republicans like to suppress the vote and Democrats don’t. I’ve long suspected this story to be false. The real story is that basically everyone likes to suppress the votes of people they don’t like and abhors suppressing the votes of people they do like. This is not to say that everyone would endorse literally any measure to suppress disfavored voters, including explicit laws categorically suppressing entire groups from voting. It’s just to say that most would endorse measures that reduce the voter turnout of their opponents, provided those measures have at least some kind of halfway plausible-sounding justification."
Arrow's Impossibility theorem says that it is impossible to design a "fair" election system, but one principle that we would all agree on, at least in the abstract, is the principle of "one person, one vote." Given that, clearly regulations which aim to prevent people from voting are wrong.
Primaries have always harbored an awkward relationship with this abstract principle. The least suppressive rules might be to allow everyone to vote in every primary, but if we did this, then Hillary Clinton would have won both the Democratic and Republican primaries! Obviously, excluding some voters is the whole intention of having party primaries.
Over the past decade or so, states have dramatically opened up their primaries to help enfranchise more people. Not long ago nearly every state was more restrictive than New York is now. As I recall, when I registered to vote, Ohio primaries were totally closed—you had to declare your party when you registered to vote, and getting that changed was kind of a pain. Incidentally, in the time between when I registered and my first ballot, Ohio changed the rules and made party registration automatic upon voting in a primary, effectively allowing anyone to show up and vote in whichever party they choose. Other states followed suit, some going even further and others stopping somewhat shorter. All of this liberalizing of the primary rules followed from a growing ease among party insiders, who thought that more open primaries would help elect moderates with more cross-party appeal since supposedly hardcore party insiders wouldn't be the only ones at the polls anymore.
2016 has upset the conventional wisdom on open primaries. There's now a strong sense that the openness they've cultivated over the past decade have enabled Donald Trump and Sanders—two radicals who are not only not actually members of their respective parties but who harbor views much more extreme than their parties' platforms.
In response, I expect most states will now move to slowly claw back some of the new open-primary rules.
"Clinton was always up in the pre-election polling". Pre-election polling was registered dems/likely voters, did not include independents. Of course, real counterfactual needs to adjust for things like strategy and advertising which would've been different. But still, had new york been open -- all else equal -- would've been much closer.
No, I don't think it is symmetric. Republicans are wealthier and more motivated. Increased turnout favors Democrats. Not that they would object if it were possible (eg ban absentees?) but generally isn't. Increasing turnout is what works for Democrats (pay people to vote?). I do expect them to tamp down though that would be a shame. It's not like New York was open, and Trump is actually the more moderate.