To add an additional chime for the basement, I am going to assume that you have a front and back door which ring your chime either once or twice depending on location. With that said..........
In that case the door chime will have 3 wires connected to it
1 front
1 back
1 transformer

run a 3wire 18 gauge from the original chime to the new chime. Splice the 3 wires on to the existing terminal screws at the old chime, and terminate the 3 wires in your basement at the new chime. Keep the wires straight as to the front , back doors and transformer, dont mix those up.

If you cant get to the old chime, you can splice the three wires to the original three wires if they are accessable in your basement

There is an add on available that does not require running additional wire. You simply connect a transmitter to the existing chime and plug in the auxillary door chime in your basement. That is what I did because I have the same issue as you. My chime was purchased from ******* for 20 dollars.