20-Year Prospect
Bike messengers will likely be around for quite some time, though the jobs will dwindle as electronic delivery becomes increasingly popular in professional and legal settings (source). However, items like food and purchased goods, which can't be transferred digitally, will still need delivery. As long as people need physical, tangible items (and need them ASAP), then messengers will be around. It's up to you and the company you work for to adapt to the changing industry (source).
Who knows: Perhaps someday you'll start your own fleet of bike messengers who hand-deliver fresh groceries for each nightly meal, or who rush-order the latest tech device to celebrities who'd rather not wait in line and who need the latest gadget, like, yesterday.
Until we invent teleportation or are speeding around in pneumatic tubes, this job will still be around in some form or another. But in all honesty, you probably won't want to be doing it in twenty years due to the immense toll it takes on your body. This is a great gig to do part time, but isn't the best long-term commitment.