You can pretty much 'bit bang' (write software to sent out the bits) 38.4kbps on any common microcontrollers these days. Additionally, all but the 'smallest' (feature set-wise) microcontrollers come with a hardware serial port as well. You might as well chose an Atmel AVR, a TI MSP430, or a Microchip PIC -- all of these are easily available, have tons of resources freely found on the Internet, etc. PICs have been around longer so there's no problem finding lots of proto boards and help, AVRs have a nicer architecture and are almost as popular, whereas MSP430s are a little more nichey and therefore not quite as prevalent.
For what you want to do, even a (PIC-based) BASIC stamp would work fine and you could be finished with your project in an evening.
---Joel Kolstad