Like most fields, whether or not a given career path is stable depends on a number of factors, many outside of our control.
What's neat about graphic design though, is that the core skills you'll learn—critical thinking, problem solving, visual communication, craftsmanship—are all things that can help you across many disciplines, even outside of the design world.
As a working designer, I've found that this field offers a lot of opportunity and flexibility depending on where your interests lie. Web design, illustration, user experience design, brand design, and more are all potential career paths you can explore within this industry. That's something I've always found exciting about design—all the different places it can take you.
I hope this helps! Let me know if you have any other questions—happy to lend a hand.
Yes. Graphic Design can be a stable career, in that you have a steady job and make a decent living. My sister in law just retired after 30+ years as a graphic designer for a utilities company, and she made a good living and really loved her job. Graphic designers work for companies, or as freelance designers.
There are many opportunities for graphic designers, because graphic design is so important in our world--everything from your toothpaste tube, to your cereal box, to the websites you look at, to the magazines you read, to the signs on the stores where you shop, to the logos for your favorite fashion brand, all rely on the talents and skills of graphic designers.
You've already gotten some good advice already. Getting a full time steady graphic design job can be competitive, not everyone gets the perfect job right out of school. Many young graphic designers have to have a second job to help them make ends meet, until they improve their skills and land a steady job. Be prepared to work hard in your art classes in school, listen to your art teachers, work on your portfolio, stay current with the latest computer graphic technology, get a graphic design internship during your summers, be humble and ready to learn from more experienced designers as you progress, and be persistent.
"The projected change in employment of graphic designers over the next ten years varies by industry. For example, employment of graphic designers in newspaper, periodical, book, and directory publishers is projected to decline 22 percent over the next ten years. However, employment of graphic designers in computer systems design and related services is projected to grow 20 percent over the same period. Companies are continuing to increase their digital presence, requiring graphic designers to help create visually appealing and effective layouts of websites."
So if you combine your Graphic Design skills with Web Design or similar skills, you're chances of finding work and advancing are higher.
AIGA (Association for Professional Design) also has a career guide for Graphic Designers: