Obama should be president because he won the vote and qualifies for the job. There have been some noises that he does not qualify and some challenges regarding his birth and citizenship but these allegations have never been backed up with any credible evidence and should be stored in a vault with most of the conspirator theories that abound with almost any political figure or event.
Yes, Obama should be president, because he was voted in as president. He is going to go down in history as one of the worst presidents this nation has ever seen, but he was voted in. TWICE. Now everyone who voted for him (and those of us who were just dragged along for the ride) are having to face the consequences of that decision.
If this question is asking whether or not he should be our current president, I say the answer is no. He still had 4 years to prove himself worthy, make things better, and offer "change", but ultimately, nothing good really came from his first term in office. I think so far he has failed.