In a wide open field I'd give it to the Klingons. Spartan strength was based on holding formation. If the Klingons can cause them to break formation, I think Klingons have better innate physicals and at least equivalent combat skills for 1vs1.
Quite interesting topic. :) Klingons have physical advantage - for example: when Worf was only thirteen he accidentally killed human boy during a soccer match, their heads had collided when they went up for the ball, Worf didn't even feel the impact, but he broke boy's neck. Klingons also have extra organs, this allowed them to survive severe injuries in battle and give them greater stamina on the battlefield. They had twenty-three ribs, two livers, an eight chambered heart, three lungs, and even redundant neural function and multiple stomachs. Klingons also have exoskeleton that protects vital organs. Standard bat'leth is made from metal called Baakonite - there's very little info about this metal, but considering the fact that Baakonite is used by advanced, space civilization I would say that it is far more resilient than weapons made from bronze. On the other hand Spartans have shields and that give them proper protection when Klingons are exposed. Also Spartans have range advantage - the hoplite spear was 9 feet long, when traditional bat'leth was typically approximately 116 centimeters long. We can't also forget about phalanx formation - this formation give Lacedaemonians great protection. In the end it all comes down to tactics - can Klingons break trough a phalanx formations? Sadly, I don't think so. Taking into account range advantage and better protection I would give the small majority to Spartans, however considering that Klingons probably have better weapons and they are definitely both stronger and more resilient, I would say that Spartans can only win six out of ten encounters.