Until yesterday, corporations could speak directly on issues. Corporations could also set up PACs which individual shareholders and employees could contribute to, and that PAC could advocate for or against candidates. But up until yesterday, corporations could not use their general revenues to elect or defeat a candidate. Those revenues are far, far greater in magnitude than what PACs could raise before.
This effectively gives the officers and directors of large corporations the ability to use their shareholders' money in a way they may not be aware of or even agree with in order to determine the outcome of specific elections, changing the makeup of specific Congressional committees, say. (You could argue that corporate governance would limit this practice, but we don't know for sure yet. It's a brave new world.)
Worse from the perspective of about half the population, these corporate expenditures will be made overwhelmingly in favor of one party, the GOP, because the CEO class is ideologically conservative. The conservatives on the supreme court have once again put a thumb on the scales to help their party, just as they did in the 2000 election.
I don't think a constitutional amendment to make the Free Speech clause inapplicable to for-profit corporations is either realistic or worthwhile. But an interesting question would be whether state corporate laws could limit the powers of businesses from making expenditures advocating for the election or defeat of any candidate. Since, presumably, each state could abolish their corporate laws and exclude corporations from their borders, I would think they would also have the authority to take this lesser step of limiting the things they may do. However, I don't put it past the conservatives on the Supreme Court to decide there is a "natural right" to incorporate, since they seem totally results-oriented.