Seaweed (and fish) has been used for both compost and mulch by people who live near the ocean for millenia. About ten years ago my neighbor spent an entire day bringing up tractor loads of it to compost for his garden. In New Brunswick in the early 19th century, Acadians planted their potatoes between two herrings. In fact, their herring fishery existed almost entirely to supply the compost they required for their fields and gardens. I've always wanted to try the herring method.
If you google seaweed as mulch, there are some good articles on it. I'll advise you that it does shrink a lot.
Most people say the salt content of seaweed is actually minimal enough that it doesn't do any damage, and that rain water tends to have washed it out if you're getting your seaweed from up the beach. Collecting seaweed that's been washed and rained on a few times means less salt, less damage to the environment, and that some of the composting has already been done for you. It's also usually a little drier, so you won't have sea-water running all over the place as you take it home. Others add that if you're worried about it, you can simply lay it out in your driveway and rinse it off with the hose.