Pre-1900 (Historic): You all need to have a closer look at John Breckenridge; he would never have sat idly by as Buchanan did and allowed the Union to disintegrate - he actually had a bit of Lincoln in him in that regard (they came from approximately the same area and were related by marriage). Perhaps the ultimate irony, Breck was the States-Rights, pro-Slavery candidate in 1860 while he privately opposed the expansion of slavery.
Breck actually was a Unionist from a border state who did not join the Confederate cause until he saw what he perceived as Lincoln usurping the Constitution, and "willful destruction" of the Union by his actions. He was expelled from the Senate in 1861 for denouncing Lincoln's Administration on that very issue. His enemies denounced him as being pro-Southern at the time of his expulsion from the Senate, and he had some real concerns that he might be arrested by the Union authorities for sedition.
He decided to join the Confederates not because he was from a Confederate State, but because he saw what Lincoln was doing as corrupting the Constitution. In doing so he ran a very serious risk of being hung as a traitor (for unlike Lee, Davis or Jackson he was what we would call an actual defector).
It's hard to chose one as worst from this period, because apart from those who did become President in the early years of the Republic, they were a pretty mediocre lot -- including Tyler, Filmore, Johnson and Arthur. Breckenridge was actually a step above this crowd.
Andrew Johnson is, IMO, the worst case scenario lived out for this period of the Presidency. Of course if he had been removed and Ben Wade had become acting President, now that could have been a dark chapter too.
Post 1900 (Modern): #1 Spiro Agnew. I've got a time-line on that point; he was an extremely weak and ill-informed person; highly susceptible to being influenced by others and yes, he was a crook long before he became Vice President.
# 2 Dick Cheney. Richard III without the page-boy hair cut. One of the most dangerous persons ever to hold high public office, more dangerous than Nixon on many levels.
# 3 Dan Quayle. Sarah Palin in an empty suit. Like Agnew, too easily influenced by others because his own level of understanding was limited.
# 4 Richard Nixon. Wasn't ready to be President yet in the 1950's. His experiences in the 1960's made him a better President (and dragged him down, the proverbial double-edged sword).
It's not clear what the Vice Presidency was ever supposed to be. Reading the Constitution, it kind of looks like a spare official who could run things if needed. Going by the original design of the Electoral College it looks like it might have been second prize in the election - a sort of consolation prize for the # 2 candidate, but if that was ever the case, it didn't work out that way.
The political role developed with the political parties; but prior to 1945 Presidents were not given to using their Vice Presidents too much even in this role, except during elections. They were simply put, unimportant and, as Daniel Webster observed, men who were buried before they were dead.
Harry Truman was the first Vice President to have a bodyguard, no one though a Vice President needed one before that point. Lincoln never actually met his first term Vice President Hannibal Hamlin until after they had both been elected to office in 1860. Franklin Pierce's Vice President William R. King died very early in their term, having never set foot in Washington as Vice President. No one missed him.