I don't see the leakage in that picture (perhaps you already cleaned it up?) but the short answer is yes, if a capacitor has leaked electrolyte onto the PCB then it can cause some damage. The electrolyte is corrosive, so it can eat away at PCB tracks etc.
Electrolyte is also conductive, so if the failed capacitor itself isn't causing any fault symptoms then the leaked electrolyte can.
In any case, leaked electrolyte should be cleaned off thoroughly, and the offending components replaced. Also look for bulging capacitors - these, if not leaking already, will eventually leak and so should be replaced as well.
Hope this helps.
AH