Flooded LAN RJ-45 connector may damage connected router?
dish water is unlikely to damage the router, but it could degrade the connections between the gold-plated contacts and the wires behind them. 

If you're worried rinse the cable with clean water, pat dry and then rinse again with denatured alcohol then dry again, that should get most of the moisture out.  The router will be slightly warmer than ambient so it should dry itself the rest of the way.