First, try disconnecting one wire from upstairs and see if the downstairs works. Sort of to validate your wiring.
They used to mount the transformer to a basement light fixture. Look in the furnace room. I wouldn't think they would seal off access to it.
An alternative. You may have bought a chime for the basement that requires more power. You could try buying a different model. If it doesn't work either, then return it.
You could install a new transformer downstairs for both bells. I assume that you have run 2 wires from the upstairs bell to the downstairs bell location. Well guess what, you could install the new transformer at the downstairs location for both bells. However, you will need three wires run between the bells. If you do this, tape the wires from the old transformer, you don't want them to short out and overheat a transformer that's hidden behind a wall somewhere. If you need help in wiring this type of solution see next paragraph.
Draw 4 boxes on a piece of paper and label as upstairs bell (UB), downstairs bell (DB), new transformer (T), and pushbutton switch (P). Then label the left side of each box a terminal #1 and right side as terminal #2. Now connect the followig: * Neutral side of the transformer T#2 run through wire1 upstairs and connect to switch P#1. * Hot side of transformer T#1 connect to DB#1 and also run through wire2 upstairs and connect to UB#1. * Now, upstairs connect other wire from the switch P#2 to UB#2 and also run through wire3 downstairs and connect to DB#2.
This probable sounds 4 times more complicated than it it. Good luck, and happy wiring.