Yes, as stable as any other job.
If you look at three of the most common kind of web development positions— product shop, consulting, and freelance— the risk of unemployment is more about the type of company, than the work itself.
Product shops come in sizes large and small. If you work for a small company, like a startup, your job security is volatile. The company could run out of money, shut down, or be sold and reorganized. But this risk applies to anybody working at such a company, including product managers, designers, copywriters, marketers, human resources, etc. And it applies to anybody working at any small business.
Consulting job security relies on a number of factors, including venture funding and market demand. Again, these are not factors that just affect developers.
Freelance work is unstable by its very nature. You have a contract, and you keep that contract with some luck and by doing good work and meeting client expectations. Sometimes you don’t have a contract.
From my experience, developers are much more likely to leave a company on their own out of boredom or shifting personal interests than to be let go or fired.
On the whole, web development and tech jobs in general are predicted to grow in the next decade. If you are risk adverse, look for a job at a large, established product shop. Most importantly, keep developing your hard and soft skills and you will find ample web development opportunities wherever you go.