62-year-old developer here. I was at a small company until it lost funding and went under five years ago. At the age of 57, I was freaking out about finding a new job. Three weeks later, I was hired at my current company. Obviously this is a single data point, an anecdote, etc. However, I have seen the same scenario for several friends my age.
The place I work at needs to find and hire exceptionally talented people (embedded software in life-critical medical devices; sloppy or buggy code, someone could die.) It knows it does not have the allure of big names such as Google. So, it does something like the Oakland A's, and plays Money Ball. It goes for the pudgy, dumpy, overlooked players who always seem to get on base somehow. It knows it has no chance at the golden children, the 27-year-old Stanford PhD's who are perfect in every way. So, it casts a wide net, and looks closely at people who might be overlooked by other organizations.
I managed to luck into finding a great, well-connected recruiter. He was instrumental in helping me get placed quickly. And, as you've heard so many times you will probably scream if you hear it once more, "networking". A friend who already had a job at my new company put in a good word when I was going through the recruiting process.
I heard a great line about mathematicians: "Mathematics is a terrible profession. The only people who should do it are people who can't not do it." There are people out there who feel that way about programming. My daughter teases me that when I retire she knows exactly what I will do: spend more time hacking on open source projects.
The core truth of your inner being at some point overtakes you. If you live and breathe to code, if you still secretly wonder why people actually pay you to play all day with computers, you will be fine as you get older continuing to be a programmer.
One thing about getting older: You begin to realize, "Now is the time." No more resume padding, no more attempts at strategizing to optimize career moves down the road. No more doing things you hate in order to lay the groundwork for that ineffable special something that seems to beckon from just over the horizon.
Older people don't need to be told, "Be honest with yourself, find your passion" etc. Been there, done that. Life will do that to you. Expect that when you are in your 50's or 60's, you will have settled in to what really does work for you. For some, it is software development. For others, it is full-time church work. (My wife.) For others, it is gardening. (A retired former executive I wave to every morning as I'm heading out to work.)
Bottom line: You would not want to work at a place that is too arrogant and stupid to consider older job candidates, or that turns its nose up at anyone who does not fit a preconceived template, or that looks to monetize the naivete of young developers. Their loss. They are doing you a favor by passing you over. Screw them.