Most of the dive shops will have secure storage for your equipment. the public buses are fine just watch what you leave overhead inside. Either have it directly under your feet or stored under the bus with secure tickets. THe best bet if you want to be more secure is to take interbus. Check them out on the web. They are very secure hotel-hotel travel. (Or hostel!). I know that the hostels in San Jose offer storage whilst you travel around and this may be the case for some of the other hostesl. Just for info i had left bags previously at Tranquillo backpackers. Once a bag went missing and they subsequently told me that one of their employees had been taking bags but had since been fired. Since then (3 years ago) there haven't been any problems. As a dive shop we offer storage no matter where you are staying.
STARTPAGE. If you are getting off the beaten track i would store it so that you can travel a bit lighter. If you give me an idea where you want to go in Costa Rica i can get you some more info.