If you Google "South Padre Island", it seems like 99% of what you will come up with is about spring break. I haven't been myself, but it seems pretty crazy. However, the rest of the year seems to be quite calm and actually has a lot to offer for such a small island.
Things to do:
1. Enjoy the beach. Yes, this is obvious, but there are a variety of ways to do this. There are parks you can park at and walk to the beach, or you can drive your car onto the beach.
2. Fish from the pier at Isla Blanca park. Here the water is quite calm and you can saltwater fish. Supposedly you can also snorkel here, but watch out for fishers!
3. Visit the Port Isabel Lighthouse. For a small fee you can climb to the top and get a view of historic Port Isabel and the coast.
4. Windsurf/parasail. Often South Padre has strong enough winds for these activities, and the water is warm enough (even without a wetsuit) most of the year.