I am a blind PHP developer. The way I program is, I'd say, different than the ways mentioned in the other answers. Before I go on, let me tell you a bit about myself.
I have been blind since birth. My eye condition isn't really important, but the point is, I have never had the problem of "losing vision"; it hasn't been there to start with. That makes a lot of things much easier.
I personally use an IDE. A lot of blind people don't, and that is mostly because their interfaces are not fully accessible to the screen readers we use. One of the problems with competition is that people are too busy caring for the majority to actually listen to the minority. For example, the whole suite of Jetbrains IDEs are not accessible. You can see the issue here: Accessibility For The Blind And Visually Impaired : IDEA-111425
I personally use Zend Studio, which is based on Eclipse, the IDE most of you probably know and have used. Eclipse is one of those exceptions that usually brings a smile to your lips; they have implemented accessibility very nicely. The whole thing isn't completely accessible, but I can live with 80%. Being blind, you learn to live with what you've got.
So, this eliminates having to memorize method signatures, documentation, and a whole lot of other things, effectively leaving my brain to worry about other things – like why legacy codebases suck so much.
I don't use a braille keyboard. Having only six keys that you have to press simultaneously to type every character is a lot slower than having a lot of more keys that you can press quickly, one after the other. I also do not use a braille display; I just use a screen reader set to 420 words per minute, which is a lot faster than the speed one would read at while using a braille display.
The most important thing, I believe, is to keep yourself ahead of your competition. When you are blind, that is already difficult because you don't have one of the senses that a lot of the people in the world have. That wouldn't be much of a problem, except what I previously mentioned about majority versus the minority; since you're in the minority, you have to find ways around things a lot of people take for granted. Therein, I believe, lies the strength that comes with being disabled: you learn to improvise, to push on, and to achieve the impossible every day merely to prevent yourself from going jobless.