Is the activity of enzymes in the human body affected by the outer temperature?
Enzymes usualy have a relative narrow temperature optimum, for those of our body this is usually around 37°C. It is around 37,2°C in the morning and goes slightly up to 37,7°C in the evening (see reference 1 for details). The temparature optimum for most enzymes looks somewhat like displayed in the figure (from here, interesting to read):



Enzymes are relatively tolerant to lower temperatures (although their reaction rate is much lower), but generally they work. If you go too high with the temperature, the activity will soon be reduced to zero, as the enzymes are denaturated and are not able to catalyze any reaction.

The human body does quite a lot to control its temperature and to keep it within a pretty small limit. It does either excessive sweating (when it's hot) or shivering (when it's cold) to maintain its temperature within or at least near the optimum. For further details on the themorgulation have a look on the Wikipedia article on this topic or reference 2 and 3.

Although our enzymes work at slightly lower reaction rates when the body temperature is higher (think of fever), this is not a major issue. Slightly higher temperatures can thus be tolerated, although you don't feel very nice. 

The effect of to hot temperatures is different, though. It is mostly due to not consuming enough water, so that the body has not the possibility to control its temperature via sweating. This condition will ultmatively lead to a heatstroke, see here for more details. Before you reach the temperature in which your enzymes reaction rate will go down drastically, you have long collapsed.



References:


A critical appraisal of 98.6 degrees F, the upper limit of the
normal body temperature, and other legacies of Carl Reinhold August
Wunderlich.
Skin blood flow in adult human thermoregulation: how it works, when
it does not, and why.
Thermoregulation: some concepts have changed. Functional
architecture of the thermoregulatory system