The guard serves a couple of purposes, and there are both in-universe and out-of-universe reasons for it to be present. In fact, these reasons are actually somewhat intertwined.
Out of universe, we have the fact that Count Dooku's style of fighting is influenced by real-world fencing, because the character was played by Christopher Lee, who was a fencer himself. The curved and guarded lightsaber hilt was designed to be suggestive of this style. In the original trilogy, Obi-Wan Kenobi, Darth Vader, and Luke Skywalker had used a quite different fighting style. Although their weapons were called "lightsabers," they all weilded their weapons more like two-handed broadswords. Playing to Lee's skill, Dooku was given a fighting style more like sport fencing, and his lightsaber hilt was curved, just like the hilts of many fencing weapons or battle sabers. The inclusion of the guard was another gesture in this direction. Compare it to a typical design for a curved battle saber.
It was common for sharp-bladed sabers like this to have both curving handles and partial guards to prevent the wielder's fingers from slipping up onto the blade. So beside the visual similarity that is thematic from a storytelling perspective, the blade guard on the lightsaber plays the same role in universe as a real-world saber guard, keeping Count Dooku's fingers safe.
Moreover, since lightsabers are made by their wielders, we also get a picture of Dooku's character at bit from his weapon. He deems himself a master of blade skill, and gave himself a curve-hilted fencing weapon.