In a two party dominated system, this would be a good way to wrest control of our political policies and discourse away from the fringe extremists and more towards the majority in the center.
We agree on more issues than we disagree, but that is not refleced in our elected officials who are elected by extremists in the primaries and by party lines in the general. It would also allow more third party candidates to have a chance to be elected.
If citizens want politicians to reflect the will of their constituents, this would be a huge step in that direction.
Edit: For clarity, an open primary is where voters are not forced to declare a party. If one wants to vote in either the Democrat or Republican primary, one can do so no matter their affiliation.
By - AgaricX