This is the CDC's travel advisor site for Thailand: http://wwwnc.cdc.gov/travel/destinations/thailand.htm
Their advice on areas with malaria Malaria, which echoes the advice here:
Areas of Thailand with Malaria: Rural, forested areas that border
Burma (Myanmar), Cambodia, and Laos. Rural, forested areas in
districts of Phang Nga and Phuket. None in the cities of Bangkok,
Chiang Mai, Chiang Rai, Koh Phangan, Koh Samui, Pattaya, Phang Nga,
and Phuket. (Updated September 2, 2011)
While some strains of malaria in Thailand are resistant to treatment, an experienced travel doctor should be able to figure out what treatments they are not yet resistant to. The CDC's map of Mefloquine-resistance is below:
Their recommended treatments are Atovaquone-proguanil or doxycycline. Be aware that Doxy, while dirt cheap and effective, does give you a sensitivity to sun exposure, so if you're anticipating spending a lot of time on the beach with exposed skin, etc. it's probably not the best bet.
Make sure other immunizations you have are up to date for your kids - things like measles are quite prevalent at times, and other more "exotic" vaccines may be suggested by the CDC or your travel doctor - Hepatitis A and B, Typhoid, etc.
Dengue Fever is probably a bigger deal, as it is an actively circulating disease in all parts of Thailand. Unfortunately, there is no vaccine against Dengue, as it's an extremely complex disease to design a vaccine for, so insect repellant, loose, covering clothing and other measures to protect against bites are the only ways to means of preventing infection.