In light of the recent investigation of Hillary Clinton and her aides in regards to E-mails sent via her personal E-mail that contained classified material, I am wondering if the state department has the power to deny the President of the United States security clearance?
No. Classified information exists pursuant to executive order (specifically, EO13526 at the moment). Congress has passed some laws to affect some portions of that (such as banning clearances for people who are actively using illegal drugs), but the laws give extreme discretion to the President. The President can, with a stroke of his pen, declassify literally every single piece of classified information that isn't a nuclear secret (those are classified pursuant to a different, statutory regime set out in the Atomic Energy Act). He can demand to see any piece of classified information held by the US government; for people in many senior posts, he can fire them on the spot if they try to stonewall.
But that's not the only reason. The President is a constitutional officer. His inherent duties, and the qualifications required for the office, are laid out in the Constitution. No law can modify those, and neither can an executive order. For the same reason, neither members of Congress nor federal judges must be approved for access to classified information when it's necessary for them to carry out their constitutional duties. Neither the President, nor members of Congress, nor federal judges can be required to keep and maintain a clearance to do their jobs. Even an act of Congress couldn't make the President have a clearance.
That doesn't mean a member of Congress or a judge can just demand to see some piece of classified information; they're not exempt from need-to-know (access to classified information requires both a clearance and need-to-know that information), and if they and the executive branch disagree on need-to-know it can be resolved like any other interbranch conflict. But the President is the head of the executive branch, so if he and an executive branch official disagree on need-to-know, the President wins (it's less "checks and balances" and more "the boss can tell his subordinates what to do").
Certain acts of Congress might restrict some information from going to the President for privacy reasons (although not always; for instance, while tax returns are private, Congress decided that the President can personally request any tax return he wants for any reason and just has to disclose that to the appropriate congressional committee), but nothing impedes his ability to carry out his constitutional duties in national security.