#1 The Worst: Lyndon Baines Johnson
One of the reasons that John F. Kennedy looks pretty good as a foreign policy president is because of how bad the foreign policy performance was of the man who replaced him. Say what you will about Lyndon Johnson's domestic accomplishments. They are impressive. As steward of America's global role, Johnson was a train wreck.
The obvious answer as to why LBJ was so bad is, of course, Vietnam. But it was how he screwed up the war that really explains his terribleness. First, his foreign policy thinking was defined by a knee-jerk misuse of historical analogy (Munich, appeasement etc). Second, a focus on the domestic implications of foreign policy decision-making led him to adopt maximalist positions - and made him deeply fearful of any policy shift that would lead to charges of "retreat." (Johnson was deeply affected by the "Who Lost China" attacks of the 1950s and was apprehensive that he would be attacked in a similar manner if he lost Vietnam.) Third, he derided principled war opposition as unpatriotic (in 1966 he called anti-war advocates "Nervous Nellies") and escalated the conflict surreptitiously, without fully informing the American people. The credibility gap he created on the war went a long way toward undermining the confidence of the American people in their elected leaders.
Lastly, as the country became more enmeshed in the war he was practically immune to information and opinion that contradicted his biases. He surrounded himself with supplicants like Walt Rostow who told him what he wanted to hear and got rid of those who offered a dissenting view (for example, Secretary of Defense Robert McNamara).
By the fall of 1967, Johnson had one last opportunity to save his presidency and end the war. Instead he continued to mislead American people about how Vietnam was going and the nature of the enemy threat - an effort capped off by General William Westmoreland's famous 'light at the end of the tunnel' speech. The result was that he hopelessly divided the Democratic Party (a rift that has never completely healed), destroyed his domestic agenda, cost himself a chance at re-election and helped to ensure Nixon's victory in November 1968. Put it all together and you have a President whose foreign policy was one for the ages - and not in a good way.
A Category All His Own: George W. Bush
The only saving grace that keeps Lyndon Johnson out of the cellar is that this list included 21st century presidents as well. After all there are bad foreign policy presidents . . . and then there is George W. Bush. The decision alone to go to war in Iraq, an unnecessary and pointless conflict based on dubious intelligence and hyped threats of nuclear, biological attack, would place Bush squarely in the cellar. Yet there is so much more. In making the case for war, Bush ran roughshod over the international system that his own country had helped to form; angered and undermined key allies and cratered the US image in the world. A failure to prioritize post-war planning ensured that a successful military victory against Saddam Hussein turned into a long-term and disastrous occupation that weakened America even further. It's pretty hard to fight a war that literally does not one thing to further US interests and strengthens the enemy you nominally went to war against in the first place (al Qaeda), but Bush accomplished that feat.
Iraq also diverted necessary resources from the war in Afghanistan and the fight against al Qaeda (leading to further escalation and American loss of life after Bush left office). Bush's second term Freedom doctrine to spread democracy around the world failed badly and opened up the US to charges of hypocrisy (it also led in part to a Hamas government taking over in Gaza). And Bush offered little response as North Korea officially became a member of the nuclear club during his presidency. Bush supporters will argue that it's too early to judge the success of his foreign policy performance. Perhaps, but early judgments are in; and they're not good. It's pretty hard to imagine any situation under which that judgment will be reversed.
*This sentence originally referred to LBJ as Kennedy's "predecessor."
**This sentence originally mistook Iran for Iraq. We regret the errors.