Parties prefer closed primaries because they keep the choice of nominee in the hands of party members. That protects the party's brand and its leadership's influence, and it shields incumbents from voters who are not loyal to the party. IVP argues this is a private interest that should not control a public election.
The Constitution protects a party's right to keep non-members out of its own affairs. IVP's answer, made in Rieckhoff v. Berger, is that it equally protects a voter's right not to join, and that a state cannot make joining a party the price of a ballot it pays for.