Good timing, you can only control 12 Servos from an Arduino UNO using the standard servo library, however you can control 20 using only 4 pins with this library that I published yesterday -http://rcarduino.blogspot.com/2012/10/arduino-serial-servos-20-servos-4-pins.html
Notice how that i used the term 'control', thats because Arduino is easily able to send the control signals for your servos, but it is not a power supply, it will not power even one servo reliably.
This is not a problem with Arduino, its how servos work, they expect a control signal and a power supply, they do not expect or need them to both come from the same device.
The power supply for your project is going to be the biggest cost, you will need strong batteries, LIPO batteries are a good option, they are light which is going to be important for a walker if its supporting its own weight.
As well as the batteries you will need a charger for them and one or more regulators to give your servos a steady input voltage.
Look at whats involved in sourcing this lot and then worry about the code.