Umbran said:
Both my karate instructors and my local police agree - martial arts classes and self-defense classes are not the same thing. They teach very different habits and skills, for different uses.
You might want to ask at your local police station - they often run free or heavily subsidized self-defense training, that may teach you more in what you need to know as a jogger on the street more quickly.
I think that it depends on what you're taking. Some martial arts (assuming karate is being used as martial-arts shorthand) are very pragmatic and focus on self-defense, while others focus on artistic expression or general fitness or tournament competition.
The school I practiced at had classes on Tuesdays and Thursdays. Both classes had a good fitness section, but on Tuesdays, after you worked out, you did self-defense -- people attacking in spontaneous ways, and you learning to do the techniques when real-world stuff got in the way and it couldn't be pretty. On Thursdays, you did sparring and general-fitness. (And in private classes, you learned the actual techniques you needed to learn to advance.)
The disadvantage of martial arts practice is that you have to find a school that emphasizes self-defense, and that's not always easy to spot.
The advantage of martial arts practice is that, unlike the self-defense courses, you're not going to a few seminars. You're training for years to get certain movements patterned as instinctive responses, so that the automatic "freeze" response gets overwritten by a "hit" response. Self-defense seminars can teach that, but it doesn't necessarily stick for everyone after one weekend.
For this case, I second pretty much all the advice given here. The only thing I'd add is possibly stupid, and I'm completely open to disagreement.
Wear your keys in a pocket or strapped to your wrist or whatever, and carry a pair of small hand-weights. This is good for adding a bit to the workout, and you can do curls and whatever while you walk. They don't have to be heavy -- the little 2.5-pounders are just fine.
If someone is following you, do what people are suggesting -- turn, make eye contact, smile, say "Hi." If they appear threatening, cross to the other side of the street. If they move to follow you, and you feel threatened, throw a weight at them and run. (Bearing in mind Harmon's post -- get to a point where you feel you could comfortably claim that you felt threatened, and that if police question you later, you can say that you tried to move away and were followed.)
The weight isn't anything you'll miss (unlike your keys), and most people flinch when someone throws something at them. Ideally, that gives you about half a second to start running (the goal isn't to injure anyone, it's to get them to flinch and do the "gak, move, what?" dance for a crucial second, while you get the heck out of there). Add in a whistle that you can blow on the exhales while running, and nobody in their right mind is going to hassle you -- or catch you.