

Open the vector logo in Illustrator.
Select the logo.

Make a duplicate copy of the logo by holding “option/alt” AND “shift” keys, drag down, and release. Using this method allows you to keep the duplicate copy aligned exactly to the original otherwise you have to re-align the two logos..

Reflect the duplicated logo by right clicking (control + click: Mac) and choosing Transform > Reflect.
Now, select the Horizontal axis.
You will have the following result.

Move the reflected logo up to meet the bottom of the original. Leave a very small gap in between the two. It adds a little extra dimension.
Choose the original logo and “lock” it.

Make a rectangle on top that covers the entire path of the reflected logo. We are going to create an “Opacity Mask” out of this shape to control the appearance of the reflected logo. Opacity masks are similar to clipping masks in Illustrator, except they control transparency instead of visibility.

Fill that rectangle with a White-to-Black Gradient, and set the angle to -90. Opacity masks are always made using a White-to-Black or Black-to-White gradient, White being visible, and Black invisible.

Make some adjustments to the gradient to get the desired effect out of the Opacity Mask.
Choose both the reflected logo, and the gradient rectangle by hitting Control + A: PC, or Command + A: Mac.
Apply the transparency mask by clicking on the arrow in the upper right of the Transparency Palette. Note: There is no right click for this, it must be accessed through the Transparency palette.

Now the gradient filled rectangle is controlling the opacity of the reflected logo.
To make the reflection a more realistic, choose the reflected logo, and take the transparency down to 40% or as you prefer.
Final result:

|
|
|
|
|