Ric,
The ocean in front of the river mouth is doable - but this weekend there will be lots of kids families (and dogs!) on the beach in that area - so best to go along the coast in either direction a few hundred meters and there will be no-one.
It's beach for 25 k's in either direction.
PS For those interested, and on a very positive note, the WA Planning Commission looks like it's going to stop the idiot developers who were proposing to stick 5000 homes (13,000 people) on the beautiful undeveloped land south of the river (including on top that amazing dune system that runs down to the river).
Instead future residential development will be restricted to the north where the existing river mouth town of Guilderton is.
It's great to see that not all of WA's coast line near Perth is going to get concreted.
For more info (and pics of Moore River);
guildertonwa.com.au/