You can do a basic check on an electrolytic with a multimeter. Set the meter to the "ohms" range ( say X1K). You will need to remove the capacitor from the circuit board - or at least disconnect one end of it. Short the capacitor out briefly to remove any stored charge. Next connect the multimeter across the capacitor. It should initially show a short circuit, but then charge up until it shows an open circuit. (Initially shows zero ohms rising to infinite ohms.) How long it takes to charge will depend on the value of the capacitor and the ohms range you have selected on the meter. If it continually shows short circuit, try a lower ohms setting on the meter. If it continues to show short circuit, with no sign of the resistance increasing, you have probably found the issue!
Just because it passes this test doesn't mean it won't break down in circuit. However, if it doesn't pass this test, then you have probably found the fault!
If the capacitor is dead, it will need replacing with one of a similar value, but get one with a high "ripple current" rating intended for power supply use. Using cheap capacitors will lead to another failure in due course!
Also, bear in mind that it is quite possible that a capacitor failure may have caused further damage elsewhere. Most switch-mode power supplies are designed to be "failure tolerant", but there are no guarantees. Model chargers are built down to a price, not up to a specification!
Also, its possible that the fault lies somewhere else completely. But a good rule of thumb for fault finding is 1) Check the incoming power is present. If yes then 2) Check electrolytics, followed by 3) lots of head scratching and in-depth fault finding!
Have fun!
--
Pete
It has both Mains and 12V on the supply side and it still works using the 12V input but the Mains supply side has failed. There is no fuse and no obvious sign of any overheating but after that I'm afraid I'm not clever enough to find what's wrong......... Can anyone here make any suggestions other than "Buy a new one"? I'm hoping that someone here will be able to look at the pictures and say that typically it is this or that component that fails and tell me how to check it 
. This is it........
