Allowing independents to vote in a party primary effectively turns the whole thing into an open primary, since anybody who thinks they might want to vote in another party’s primary at some point in the future would simply register independent to keep their options open (I know I would).
If you want to have a say in who gets nominated, you can simply join a party. It’s not like they charge a membership fee or something, and you can still vote for other candidates in the general election if you don’t ultimately like the nominee. The only downsides I see are (1) you’ll probably get more junk mail and robocalls and (2) I guess in a way you’re reinforcing the two-party system by registering as a member of one, since it makes it look like they have more members. And in NY you have to do it several months ahead of time if you’re already registered as an independent.
