Most people get there by public bus from the northern cities of Chiang Mai and Chiang Rai. Buses are safe, inexpensive and quite frequent with various 'classes' costing between 160 - 350 baht for the five hour trip from Chiang Mai's Arcade Bus Station (via Chiang Rai) or the approximately two hour trip from Chiang Rai's two public bus stations. Book in advance if possible as the buses fill up quite quickly.
When you arrive at Mae Sai bus station (which is a little out of the way) red van-like 'passenger service vehicles' (songtaew) will take you to the main street that leads to the border point for 15 Baht (10 baht for kids).
Many people do this trip from Chiangmai (aka visa run) in a single day. You need to be back at Mae Sai in time for the last bus at 4:30 pm - and you almost certainly need to have booked it in advance other wise it'll be full.
Some tour companies in both Chiang Mai and Chiang Rai offer a day trip minibus service to the border and back for around 500 baht from Chiangmai (less from Chiang Rai). The driver will wait in Mae Sai while you cross to Burma for a couple of hours. These can be booked from various guest houses or travel agents and will pick you up at your guest house or hotel.