Repeal the 1st Amendment, or reverse the ruling of Citizens United v. FEC
It would matter what sort of influence you were interested in reducing. Corporations have little influence when it comes to direct campaign donations, since currently it illegal for corporations to donate directly to political campaigns. This leaves Political Action Committees (PACs), whose donations to political parties/multiple-candidate are a paltry sum.
$5,000 to a candidate or candidate committee for each election (primary and general elections count as separate elections);
$15,000 to a political party per year; and
$5,000 to another PAC per year.
PACs may make unlimited expenditures independently of a candidate or political party
As far as how influential spending in political campaigns is, Steve Levitt, Freakinomics co-author found that:
When a candidate doubled their spending, holding everything else constant, they only got an extra one percent of the popular vote. It’s the same if you cut your spending in half, you only lose one percent of the popular vote. So we’re talking about really large swings in campaign spending with almost trivial changes in the vote.
Supposing you wanted to eliminate that %1-2% edge, you have the issue that the Supreme Court ruled in Citizens United vs Federal Election Commission that corporations/labor unions free speech cannot be abridged by Congress if the 1st Amendment is to have any teeth.
The First Amendment prohibits Congress from fining or jailing citizens, or associations of citizens, for engaging in political speech [...] Although the First Amendment provides that “Congress shall make no law . . . abridging the freedom of speech,” §441b’s prohibition on corporate independent expenditures is an outright ban on speech, backed by criminal sanctions. It is a ban notwithstanding the fact that a PAC created by a corporation can still speak, for a PAC is a separate association from the corporation.
Repealing an amendment (like the 21st repealed the 18th) to the Constitution requires that either both houses of Congress by 2/3rds supermajority, or a constitutional convention or 2/3rds of the state legislators propose an amendment.
The Congress, whenever two thirds of both houses shall deem it necessary, shall propose amendments to this Constitution, or, on the application of the legislatures of two thirds of the several states, shall call a convention for proposing amendments [...]
Then, the amendment needs to be ratified by 2/3rds of the states
A proposed amendment becomes part of the Constitution as soon as it is ratified by three-fourths of the States (38 of 50 States).
The second route is trickier. Overturning Citizens United v. FEC would probably require that a new set of SCOTUS Justices. The same justices would likely rule the same way as their previous opinions. With a 5/4 split between concurring and dissenting opinions, you would need for Kennedy, Roberts, Scalia, Alito, or Thomas to leave the bench. SCOTUS Justices are appointed for life, so you would need either kill one or wait for for one to die.
Additionally, to bring a case to trial, you need what is called standing. You need to demonstrate to the court how you were personally harmed by the fact that a corporate PAC caused you harm. This would likely require that you would need to run for office, and show that a PACs political free speech caused you harm.