In reply to French Erick:
Difficult to be specific without knowing where the 'Local' caves are......
Being an SSSI doesn't necessarily mean 'No climbing'. Quite a few of the famous North Wales climbing sites, including e.g. Idwal and the Great Orme, are within SSSI's, and no doubt so are many more nationwide.
It is unlikely that the bouldering/climbing itself would be a problem unless disturbing e.g. nesting birds or seal or bat colonies. However there are some areas within these sites that are particularly sensitive, and where 'cleaning' routes (including e.g. simply brushing moss off holds) could damage sensitive ecosystems.
Searching the 'Find a crag' function on this site may show whether they have been climbed / 'developed' in the past, but doesn't necessarily indicate that access is allowed. The BMC regional access database would be a good next place to search for any issues, or contact SNH / look on their site.
Or just name the caves so people on here can help !