Pick the one you're best at, and polish your skills and CV till they glow. Even if SysAdmin is more stable than Programmer, great Programmer is more stable than Lousy SysAdmin.
Couple of caveats:
No job's secure unless you perform well at it
There's not a job in existence that some idiot won't try to outsource. Hell, Britain once deported children in welfare to Australia to save money.
Outsourcing is generally avoided in the area the company actually works in. An insurance company might outsource software development, a software company that does that is making itself redundant in the long term.
Outsourced programming is generally reworked locally once delivered, so there are always jobs available there.
With that said, and this is just an opinion based on limited experience:
Defence and Government jobs tend to be a bit less willing to let people go because of potential bad publicity. But they're also prone to larger than average failures because they're unwilling to admit failure.
The Financial Industry pays well but is very much "hire and fire" in its treatment of employees. You can make good money during the good times, but be prepared to live off your savings occasionally.
Big software-focused companies(Microsoft, Google, Bloomberg, Amazon etc.)work you very hard but probably offer the best pay to stability ratio, as long as you can handle the work. Same with software consultancy (eg Accenture or DeLoitte)
The Gaming industry has the worst rep for long hours, poor pay and arbitrary layoffs.
It's important to remember that stability is relative. Adequate programmers with a few years experience can generally find someone to hire them. You may simply find that you move from one job to another.