Yes, we now have a Starbucks in town and if you find yourself about to go in, don't do it. Walk one block down Duval, take a left on Eaton street and you'll find the outdoor tables and front door to "Sippin", a great local coffeehouse with comfortable sofas and chairs, internet access and friendly staff.
Another quality Coffeehouse is on the Southern (quiet) end of Duval street in the 1200 block you'll find The Coffee and Tea House a good locals hangout where you can people watch and meet the locals, no internet here.
The third of the big three is on Caroline street at Elizabeth street. Free WiFi with purchase and computer access for rent. Open air shop with outdoor seating as well.
If your looking just to get online the Library on Fleming Street at Elizabeth has free half hour accesss Tuesday thru Saturday from 10 till 5.
There are plenty of smaller internet/coffeeshops around Old Town and I hope you find a good one and let me know about it.
A "MUST SEE" for anyone looking for the "real" Key West is a little bar-restaurant called "Nobody Knows". NOT a tourist trap, but the genuine article... the Key West shrimp fleet, crowded away from Mallory dock several years ago by the tourist-trappers, is alive and well on Stock Island. And the old-time shrimpers frequent Nobody Knows, where you get $1 drafts and live local musical talent. Come as you are, because if you dress up you will look out of place with the shrimpers. Go East out of Key West on US 1, bear right onto McDonald Ave, go a few blocks until you see the funky little "bakery" on the right.(just after the "tom thumb" store) Turn right there and go down until you come to the street with the old trailers and concrete power poles. Go left and follow this dusty road around until you come to the shrimp fleet, and right there at the dock is "Nobody Knows" If you are not sure where to go just stop at the Tattoo parlor or Dion's on Stock Island and ask how to get to "Nobody Knows". All the locals know where it is. While you are there check out the cool old fishing trawlers, some of which have been going out for over 50 years.
Find the 'smaller' Captain Tony's off of Duval Street. There are two Captain Tony's in Key West. One is extremely touristy and the other is much cooler. It does not look like much from the outside, but worth it for a couple of drinks. It is apparently where Jimmy Buffet started, and where Hemingway and Tenessee Williams drank at.
At the very least it has good live music, business card all over, and of course women's bras from around the world. Who can ask for more?
My sister-in-law saw this one, had to take a photo, but no, none of us went in to see if the sign was serious...