I have lived in Bangkok for some time, so I think I can answer this question. Bangkok has very nice public transport system but it is only available in certain area of the city. Bangkok has 2 suburban Rail network BTS and MRT. both have interconnectivity and better them and they cover almost all the main tourist attraction accept Khaosan road(Which is Bangkok's Backpacker area). BTS and MRT also have interconnected to ferry with will take you to all the temple and Grand Palace.
Considering all of this and I would recommend lower Sukhumvit area (NANA, Ratchathewi, and SIAM). Hotels in this area will be little expensive. if your budget do not permit these areas you can consider upper Sukhumvit (Ekmai and Thonglor ) they are just 10 min train ride from Downtown Bangkok.