A presidential nominee is ultimately responsible for choosing a running mate, but some of Trump's campaign advisers are afraid he may not even consult or inform them about his choice.
Buried in this report from Bloomberg about Trump downplaying the need to raise money, Trump's advisers privately say they're worried that he may select someone on a whim.
Amid all the nervousness from Republican political veterans that his speeches are too spontaneous, he vowed to show discipline in the announcement of his vice-presidential pick. The campaign has privately discussed making an announcement as soon as this month to deflect from the controversy surrounding Trump University, and some advisers have worried that Trump may decide on his own to post the announcement on Twitter one night with little warning.
Even if he does give his own staff a heads up, we can safely assume his choice will not have been vetted to any meaningful degree. Trump's rivals in the Republican presidential primacy race had more than plenty of time to vet and attack Trump, but they didn't.
They didn't because they agree with what he says and what he stands for. Trump will undoubtedly select a running mate who also agrees with everything he says and stands for. With that in mind, the entire idea of Trump vetting a running mate seems like a misnomer. Does Sauron vet Saruman? Does a garbage dump vet a garbage dumpster?
While a democratic campaign will vet a potential running mate to ensure that they don't have any wildly offensive skeletons in the closet, that would not preclude a Republican; especially not someone running alongside Donald Trump.
The great wealth of material Democratic trackers and PACs can use against Trump could be doubled by an exceptionally bad running mate and I have a hard time picturing Trump selecting someone who isn't exceptionally bad. I believe Trump would intentionally select someone worse than himself if only to make himself look better.
Needless to say, you would have to be an immense idiot or immensely desperate to agree to be Trump's running mate.