Allowing fusion candidacies
Presently most states forbid two or more parties from nominating the same person for the same office in the same election. Such a candidate would be what is called a fusion candidate. Presently, fusion candidacies are allowed in only eight states: Connecticut, Delaware, Idaho, Mississippi, New York, Oregon, South Carolina, and Vermont. New Hampshire also allows it if a primary is won by a write-in candidate.
The main purpose and effect of not allowing fusion candidacies is to suppress minor parties and entrench a two-party duopoly. It has been argued that people have a First Amendment right to associate in political parties and to nominate any person eligible to hold the office, but the U.S. Supreme Court decided by 6-3 in Timmons v. Twin Cities Area New Party, 520 U.S. 351 (1997), that fusion is not a constitutionally protected civil right.
The argument often made against fusion candidacies is that it is unfair for the same candidate to appear on the same ballot for the same office more than once, and that no one is deprived of the opportunity to vote for the candidate if his name appears at least once. The counterargument is that fusion candidacy allows a minor party to unite in supporting a candidate of another party and thus avoid dividing support by nominating someone else, and perhaps throwing the election to third least desirable candidate from their standpoint. It also allows the voters to signal their policy preferences by voting for the candidate under one party label instead of another.
Party election primaries were originally sold as a reform measure, to prevent the undue influence of party bosses on the nomination of candidates by throwing the selection open to all the voters who care to vote in them even if they are not regularly active in the party. The problem with doing that is that it leaves the decision to people who don't know the candidates as well as party leaders do, and who cannot be reached by those party leaders at an affordable cost to inform them. It opens the nomination to candidates who may disagree with party leaders on issues but who have the support of monied backers who can afford to market them to voters who don't take the time to carefully inform themselves on their choices. This leads to a division between the party rank-and-file on one side and the well-funded candidates and their supporters on the other. Those holding one set of policy preferences who hold most of the party offices may nevertheless get candidates who do not represent their views, but the views of wealthy special interests and rent-seekers.
The main alternative is party nominating conventions, probably beginning with local precinct conventions, which elect delegates to county conventions, district conventions, and finally to a state convention. Some states combine a primary with a convention process, using the primary to select the delegates from the precinct convention. Primary elections are expensive, so if all or part of the costs are paid from public funds, that is a subsidy to the parties that receive those funds, and arguably an equal protection violation if some parties are given a preference for receiving such funds. While conventions may be open to voters who are not regularly involved in party activities, it at least allows the personal contact and deliberation that is not supported by the non-deliberative process of merely casting votes on election day.
An interesting alternative is the nonpartisan blanket primary, or "top-two" system, adopted, most recently and prominently, by California, and although called a "primary", is really a non-partisan general election with what was the general election being converted into a run-off election. Although sold as a reform to reduce two-party gridlock, it effectively causes every candidate to have to form his own party and get enough financial support to get on the ballot. It affords no way for the parties to advance through a convention process a meritorious candidate who is not wealthy or does not initially draw much financial support, and thus excludes candidates who are not supported by rent-seekers, including minor party candidates. That was undoubtedly the intention of the proponents, and they have effectively excluded any candidates who are not beholden to their donors or organized special interest groups.
If the aim is to reduce the public choice problem of special interests dominating the election process then unless one goes to some radical alternative like sortition, a convention system at least offers an opportunity to nominate meritorious candidates who are not owned by some rent-seeker group.
Ending straight-ticket voting
Leaders of the top two political parties like straight-ticket voting without a mixed-ticket option, so that their most popular candidate in the election, usually at the top office on the ballot, such as president, can cause lazy voters to cast a straight-ticket vote that includes him and all the other members of his party. That enables them to pack lower offices with less competent or honest candidates, more likely to serve special interests than the people. In many electoral jurisdictions 30-70% of voters will habitually vote straight-ticket, and without a mixed-ticket option for overriding the straight-ticket choice for particular offices, if there is no candidate of the preferred party who is a candidate, the vote for that office is lost, and a minor party candidate who may be the only alternative doesn't get the votes that might otherwise have come to him. Straight-ticket voting, whether offered as a ballot option or merely a habit of individually choosing only candidates of one party, is perhaps the main reason minor party candidates generally don't break out of single digits. If the option of straight-ticket voting were at least not offered as a ballot choice, it is possible that minor party or less-finally supported candidates might gain more stature or even win some elections and thus build a record for governance.
For more on this see:
Video of debate between Kathie Glass, former Libertarian candidate for Texas governor, and Dave Nalle, National Chairman of the Republican Liberty Caucus, on September 30, 2012, who discuss these three reforms: