I've created this logo in Illustrator. This doesn't have any border. But when I applied gradient to it and then applied 3D effect, it is not cool. There appears a border on one side of logo's letters. Why is it so? Also, this border doesn't come when there is solid color. I think this issue is common. Is there any solution? Or is there any different way to do this?
There are two things at work here: the default position of the light source isn't really working for your design; and Illustrator's inability to process a gradient into an Extruded shape.
The default light source is on the front of the object, a bit to the right. That will cause the bottom and right side of any 3D object in your design to be in a dark shade.
Try and change the location of your light source. When applying your 3D effect (or when editing the existing one), click the 'More Options' button to reveal the light controls:
You can drag the white dot around to change the light source's location, or use the icons below to move it to the back of the object, create another or delete one (to a minumum of one).
Besides that, Illustrator's Extrude effect can't handle an object with a gradient fill. It will make the 3D object a medium gray no matter what the gradient's colours are—besides rasterising the gradient.
I'd advice you to create the 3D effect with an object that doesn't have a gradient, then Expand the 3D effect and colour the resulting objects by hand, only then applying the gradient. That way, you can also manually edit the colours of the sides. Do keep in mind that Expanding is, by nature, destructive, disallowing you to further edit the effect after Expanding.