If your major part of the trip to Bangkok is to visit the historical tourist attractions such as Grand Palace, Wat Phra Kaew, Wat Pho, and Wat Arun, here are my recommendations for the best area to stay in Bangkok.
Stay close to the riverside
Many of the historical attractions in Bangkok are located along the Chao Phraya River. The riverside accommodations offer you not only an easy access to the attractions, but also a gorgeous river view of Bangkok.
Stay close to the BTS Skytrain
If you cannot find a riverside accommodation, another option is stay close to the BTS Skytrain. BTS Skytrain is an effective transportation mode covering major business areas in Bangkok, and connects with the Chao Phraya Express Boat at Saphan Taksin Station. So, if you stay close to the BTS Skytrain, you just take it to Saphan Taksin Station, where you can take Chao Phraya Express Boat to the attractions. See how to get around Bangkok by using BTS Skytrain, Chao Phraya Express Boat, and also MRT Subway here.
P.S. Many people choose the latter option because BTS Skytrain covers Sukhumvit Road which is the road that is full with shopping malls, markets, restaurants, pubs, and night clubs.
Hope this information helps.