Anyone who is not constitutionally ineligible (that is you are a natural-born citizen and you have not served for 2 terms) can run for President. That's democracy.
US parties don't have any screening processes for candidates; presidential hopefuls can just announce their candidacy and file with the FEC to run for President.
That being said, different states have different rules and criteria to be on the ballot since elections are conducted by states, so one has to be politically active in all states to be on the ballot. (Actual requirements varies between different states.)
There's also the primaries for both the Democratic and Republican parties which is the nomination stage for both parties. This allows voters to choose their candidate to compete in the general election. If a candidate has last scandals and they get nominated by their party, they are still chosen by voters. The voters have a choice to not vote for them.
Candidates are also under the focus of the media, most of the scandals we know are all first reported through the media. If not, most of us won't even know that the scandals exist.
parties can try to block someone from being nominated, against the voters' choice, during their respective conventions.
For the Republican Party, this can be done since convention rules are drawn up before each year's national convention and the party can theoretically modify the existing rules and lift the “binding” obligation, allowing the delegates to vote for anyone they want.
The rule change is a two-part process and Rule 16 is the relevant rule that binds delegates so should it be modified, delegates can be unbound. This was widely discussed and reported last year, especially for the Republican convention and the WSJ published a flowchart on how Trump's nomination could be stopped through the rule change.
Who decides these new rules? It’s a two-part process. Just a few days before the convention, the rules committee—which includes two national delegate representatives from each state and territory, totaling 112—will meet to hammer out any additions or subtractions to the Tampa rules to determine how the Cleveland convention will proceed. Once a majority of committee members approve the new draft, it then goes to the convention floor, where it must win support from a majority of the convention delegates.
As for the Democratic Party, there are super delegates who do have the final say in who to nominate since their votes are not bound. They can vote for someone other than the presumptive nominee and should the number of votes be enough, they can block a the presumptive nominee from clinching the party's nomination.
To conclude, the bottom line still is that we can choose not to vote for them.