If you do not rent a car you can get to Ban Krut by taxi, by train or by bus .
Taxi from the Suvarnabhumi airport to Ban Krut cost about 3500-4000 Baht and spent about 6 hour. Taxi from Suvarnabhumi airport to Bangkok train station cost about 500 baht and spent about 1 hour (depending of traffic). Timetable of trains you can find on the official website of State Railway of Thailand. Travel time is about 6 hours.
If there is no train from Bangkok, stopping at Ban Krut, you can also choose the train, which stops at Bang Saphan Yai. Bang Saphan Yai is the next train stop, approximately 15 minutes away after Ban Krut (by train). From Bang Saphan Yai you can get to Ban Krut by taxi, spending is about 500 baht.
Sometimes it is better to go from Bangkok to Hua Hin by bus and then to go from Hua hin to Ban Krut by train. Timetable of bus from Suvarnabhumi airport you can find on the website. Travel time from Bangkok to Hua Hin is about 3-4 hours.