Bangkok and Thailand in general does, not have any major health risks. There are no infectious diseases that you should actively worry about. Some vaccinations for Thailand are recommended but not mandatory. Health risks are only present in remote areas that tourists rarely visit. There are some excellent pharmacies in Bangkok. The quality of hospitals is of International standard in major tourist destinations such as Bangkok, Phuket, Chiang Mai & Pattaya.
Important vaccinations like Tetanus, Hepatitis A + B should always be up to date wherever you are traveling or not. Malaria tablets are not required for a trip to Bangkok or anywhere else in Thailand. Make sure you bring mosquito repellent with you or buy some after you arrive. You should apply it to exposed legs and arms if you are outside after dark. Mosquito bites are not usually dangerous but can be irritating. Local brands of mosquito repellent work very well despite what you might read elsewhere.
Travel Insurance for Thailand
You should make sure that your travel insurance has adequate health/accident coverage as hospital bills can be expensive if you have any medical issues. One of the most common causes of injury to tourists is motorcycle accidents.