Stayed at the Hotel Baltico a couple of years back. Right by the harbour. They don't advertise it on their website but they have a little secure courtyard area you can leave your bike in. Room for about 4 or 5 bikes max.
Beautiful beach a 5 min walk away, and the fisherman's cemetery overlooking the sea is amazing. Just walking up/down the steep sided bowl that is Luarca is a great way to pass an afternoon.
It's on booking.com