I looked up mixed (aka semi-closed) primaries and they mean people registered with a party can vote and independents, but not anyone in a third party (which would be weird if say Greens could vote in a Democratic Primary, but not Republicans).
I am ‘open’ to the idea of closed primaries in all 50 states, but only if it is easy to switch parties right on the ballot on election day. It was well talked about that NY’s closed primary really hurt Bernie Sanders (I think unfairly) as it was a long time before when you needed to switch parties or switch from being independent.
I agree with GuildF312S that most people would vote for the best candidates but if say everyone could vote in every party’s primary each election cycle, I don’t doubt there would be some strategic voting (people organizing online and agreeing that candidate A in an opposing party is the least likely to beat anybody else, so vote for A). There could be that strategy in my system too since sometimes the primary is a slam dunk in your state and your party so you could choose only to sabotage another party. But I am unwilling to consider the NY solution to battle this.
My preferred ideal solution is to to get rid of parties altogether and have everyone who thinks they are qualified and want to run for president go through some sort of hurdle (e.g. petition gathering) and get on a ballot with maybe 20 people on it and then have no primaries and the one general election uses RCV and no Electoral College to pick a winner. I realize that is going to happen anytime soon, so I think open primaries or closed with same day party registration is what I can compromise to. (If I could make deep changes to the US, I’d want to change a lot of other things I think are stupid, like get rid of the existing Senate and replace it with 100 senators that are elected at large over the US in a proportional representation scheme - the house can stay more or less as it is but with someone besides the party in power in control of the district definition).