Listed below are the companies that I know about. I would be very interested to hear from any other guides or tour companies and I will happily add your details to the the list.
In the north of Thailand the best bet for advice on caving is probably Cave Lodge in Mae Hong Son. This is run by a very experinced Aussie caver who can rent you equipment, advise on the caves to visit and arrange guides if required.
In Chiang Mai there is Chiang Mai Rock Climbing Adventures. Their website only mentions caving courses, not guided trips, which are held at Crazy Horse Buttress a few kilometres to the south-east of Chiang Mai.
Near Chiang Dao Cave in Chiang Mai the Chiang Dao Nature Resort can organise guides and transport to nearby caves such as the 2 km long Tham Ki Mi.
Nearer to Bangkok is Saraburi Adventure. This company is Thai run, but the director speaks good English. The website only mentions trips to a single long stream cave in Saraburi.
In the other Thai caving areas of Kanchanaburi and the south of Thailand I don't know of any companies running caving trips.