Dear Bettie,
This will depend greatly on your budget of course. If you would like a very nice serviced apartment (but at a very high price) try Frasier Suites on Sathorn Road.
A cheaper option for a serviced apartment would be Boss Tower on Rama 4 Road. A swimming pool is still included.
There is one very good website btw for finding apartments in Bangkok which is:
www.mrroomfinder.com You can even search by subway or BTS line!
Good luck
PS: Mocht een vertaling nodig zijn laat het even weten aub.